theoretical studies on magnetic and photochemical ...467777/fulltext02.pdf · theoretical studies...

Theoretical Studies on Magnetic

and Photochemical Properties of

Organic Molecules

Xing Chen

(陈星)

Doctoral Thesis in Theoretical Chemistry and Biology

School of Biotechnology

Royal Institute of Technology

Stockholm, Sweden 2011

Theoretical Studies on Photochemical and Magnetic

Properties of Conjugated Organic Molecules

Thesis for Philosophy Doctor Degree

Department of Theoretical Chemistry & Biology

School of Biotechnology

Royal Institute of Technology


c⃝ Xing Chen, 2011

pp i-xvi, 1-62

ISBN 978-91-7501-227-8

ISSN 1654-2312

TRITA-BIO Report 2012:2

Printed by Universitetsservice US-AB


献给我的父母To my parents

Abstract

The present thesis is concerned with the theoretical studies on magnetic and pho-

tochemical properties of organic molecules. The ab initio and first principles theo-

ries were employed to investigate the vibrational effects on the isotropic hyperfine

coupling constant (HFCC) known as the critical parameter in electron paramag-

netic resonance spectrum, the theoretical simulations of the vibronically resolved

molecular spectra, the photo-induced reaction mechanism of α-santonin and the

spin-forbidden reaction of triplet-state dioxygen with cofactor-free enzyme. The

theoretical predictions shed light on the interpretation of experimental observa-

tions, the understanding of reaction mechanism, and importantly the guideline

and perspective in respect of the popularized applications.

We focused on the vibrational corrections to the isotropic HFCCs of hydrogen

and carbon atoms in organic radicals. The calculations indicate that the vibra-

tional contributions induce or enhance the effect of spin polarization. A set of

rules were stated to guide experimentalist and theoretician in identification of the

contributions from the molecular vibrations to HFCCs. And the coupling of spin

density with vibrational modes in the backbone is significant and provides the

insight into the spin density transfer mechanism in organic π radicals.

The spectral characters of the intermediates in solid-state photoarrangement

of α-santonin were investigated in order to well understand the underlying exper-

imental spectra. The molecular spectra simulated with Franck-Condon principle

show that the positions of the absorption and emission bands of photosantonic

acid well match with the experimental observations and the absorption spectrum

has a vibrationally resolved character.

α-Santonin is the first found organic molecule that has the photoreaction

activities. The photorearrangement mechanism is theoretically predicted that the

low-lying excited state 1(nπ∗) undergoing an intersystem crossing process decays

to 3(ππ∗) state in the Franck-Condon region. A pathway which is favored in the

solid-state reaction requires less space and dynamic advantage on the excited-

state potential energy surface (PES). And the other pathway is predominant in

the weak polar solvent due to the thermodynamical and dynamical preferences.

Lumisantonin is a critical intermediate derived from α-santonin photoreaction.

The 3(ππ∗) state plays a key role in lumisantonin photolysis. The photolytic

pathway is in advantage of dynamics and thermodynamics on the triplet-state

PES. In contrast, the other reaction pathway is facile for pyrolysis ascribed to a

stable intermediate formed on the ground-state PES.

vi

The mechanism of the oxidation reaction involving cofactor-free enzyme and

triplet-state dioxygen were studied. The theoretical calculations show that the

charge-transfer mechanism is not a sole way to make a spin-forbidden oxidation al-

lowed. It is more likely to take place in the reactant consisting of a non-conjugated

substrate. The other mechanism involving the surface hopping between the triplet-

and singlet-state PESs via a minimum energy crossing point (MECP) without a

significant charge migration. The electronic state of MECP exhibits a mixed

characteristic of the singlet and triplet states. The enhanced conjugation of the

substrate slows down the spin-flip rate, and this step can in fact control the rate

of the reaction that a dioxygen attaches to a substrate.

Preface

All works presented in the thesis were completed in the past three years, 2008.09

– 2011.12, at the Department of Theoretical Chemistry and Biology, School of

Biotechnology, Royal Institute of Technology, Stockholm, Sweden, and Depart-

ment of Chemistry and State Key Laboratory of Physical Chemistry of Solid

Surfaces, College of Chemistry and Chemical Engineering, Xiamen University,

Xiamen, China.

List of papers included in the thesis

Paper 1. Zero-point Vibrational Corrections to Isotropic Hydrogen Hyperfine Con-

stants in Polyatomic Molecules,

X. Chen, Z. Rinkevicius, Z. Cao, K. Ruud, H. Agren, Phys. Chem.

Chem. Phys., 13, 696, 2011.

Paper 2. Vibrationally induced carbon hyperfine coupling constants: a reinter-

pretation of the McConnell relation

Z. Rinkevicius, X. Chen, H. Agren, K. Ruud, Submitted for publica-

tion.

Paper 3. Spectral Character of the Intermediate State in Solid-State Photoar-

rangement of α-Santonin

X. Chen, G. Tian, Z. Rinkevicius, O. Vahtras, H. Agren, Z. Cao, Y.

Luo, Submitted for publication.

Paper 4. Theoretical Studies on the Mechanism of α-Santonin Photo-Induced Re-

arrangement

X. Chen, Z. Rinkevicius, Y. Luo, H. Agren, Z. Cao, ChemPhysChem,

in press.

Paper 5. Role of the 3(ππ∗) State in Photolysis of Lumisantonin: Insight from ab

Initio Studies

X. Chen, Z. Rinkevicius, Y. Luo, H. Agren, Z. Cao, J. Phys. Chem.

A, 115, 7815, 2011.

Paper 6. Theoretical Studies on Reaction of Cofactor-Free Enzyme with Triplet

Oxygen Molecule

X. Chen, W. W. Zhang, R. L. Liao, O. Vahtras, Z. Rinkevicius, Y.

Zhao, Z. Cao, H. Agren, Submitted for publication.

viii

List of related papers not included in the thesis

Paper 1. Theoretical Studies on the Interactions of Cations with Diazine,

X. Chen, W. Wu, J. Zhang and Z. Cao,

Chinese J. Struct. Chem., 22, 1321, 2006.

Paper 2. Theoretical Study on the Singlet Excited State of Pterin and Its Deac-

tivation Pathway,

X. Chen, X. Xu and Z. Cao,

J.Phys.Chem. A, 111, 9255, 2007.

Paper 3. Theoretical Study on the Dual Fluorescence of 2-(4-cyanophenyl)-N,N-

dimethyl-aminoethane and Its Deactivation pathway,

X. Chen, Y. Zhao and Z. Cao,

J. Chem. Phys. 130, 144307, 2009.

Paper 4. DFT Study on Polydiacetylenes and Their Derivatives,

B. Huang, X. Chen and Z. Cao,

J. Theo. Comput. Chem., 8, 871, 2009.

Comments on my contributions to the papers included

I was responsible for the calculations, discussion and writing the first drafts of

Papers 1, 3, 4, 5, 6

I took partly responsibility for the calculations, discussion and writing of Paper

2.

Acknowledgments

How time flies! The slice of life that I was making effort to prepare my licentiate

thesis in the last year arising in my mind looks like a vivid episode that just

happened. I have spent about three years in Stockholm, and it draws to an end

by now. The feeling at this moment can not be described by any simple word. It

indeed is not a long time from the viewpoint of a whole academic career, however,

the rememberable and precious experience I have learned a lot from in these years

will be treasured up forever in my life. On the occasion of the thesis coming to

the end, I would like to express my sincere gratitude to many people, who support

me, inspire me and stand beside me always.

I truly acknowledge my supervisors Dr. Olav Vahtras and Dr. Zilvinas Rinke-

vicius. They create a free and comfortable academic environment, provide me with

a lot of help and chances to fulfil the full-skilled training and guide me to a new re-

search field. Zilvinas not only shares me with the science interests but also spreads

the optimistic living attitude over me, especially, when I am confused with diffi-

culties lying in the way. They give me the patient and insight suggestions. Many

thanks for your help.

I also genuinely acknowledge Prof. Hans Agren, a head of the department

of theoretical chemistry in KTH, for giving me a precious opportunity to study

here and a warm welcome releasing the nervous emotion in my first arrival to a

foreign country. He is concerned about my progress in my project and encourages

me even though I make a tiny achievement. I am eternally grateful to Prof.

Yi Luo for his consideration, unselfish help and positive support always. The

in-depth discussions shed light on the proceeding way of investigation and the

encouragement inspirits me to handle difficulties.

I would like to give my enormous thanks and gratitude to my supervisor Prof.

Zexing Cao in Xiamen University, who guides me to the exciting and wonderful

field of theoretical chemistry, cultivates my scientific view, and recommends me to

study in Stockholm. His rigorous working style and modest living attitude affect

me a lot.

I wish to express my gratitude to Prof. Faris Gel′mukhanov, Dr. Ying Fu,

Prof. Fahmi Himo, Dr. Marten Ahlquist, Prof. Margareta Blomberg, Dr. Yao-

quan Tu, Dr. Arul Murugan and Prof. Boris Minaev for the excellent lectures

they give. As well as I would like to thank all researchers and secretaries in our

department for their guidance and help.

x

Many thanks to Prof. Qianer Zhang, who imparts the basic but essential

principles both in science and life, Prof. Yi Zhao and Dr. Wei-Wei Zhang, who

guides me with insightful ideas, as well as Dr. Xuefei Xu, who helps me a lot at

the beginning of my academic career. The pleasant cooperations with them bring

me wonderful inspirations and great passions to do projects.

Many thanks to my colleagues, they are Guangjun Tian, Dr. Sai Duan, Dr.

Rongzhen Liao, Dr. Weijie Hua, for helping me to improve the program skills

and sharing me with good ideas in work. I am grateful Dr. Yuejie Ai, Dr. Qiong

Zhang, Xiao Cheng, Dr. Qiang Fu, Dr. Shilv Chen, Li Li, Dr. Yuping Sun,

Xiuneng Song, Yong Ma, Ying Wang, Dr. Igor Ying Zhang, Dr. Lili Lin, Quan

Miao, Dr. Feng Zhang, Xinrui Cao, Bonaman Xin Li, Dr. Xin Li, Yongfei Ji, Dr.

Kai Fu, Dr. Zhijun Ning, Zhihui Chen, Xiangjun Shang, Hongbao Li, Junfeng Li,

Ce Song, Bogdan, Staffan, Chunze Yuan, Dr. Johannes, Li Gao, Dr. Xifeng Yang,

Dr. Linqin Xue, Lijun Liang, Lu Sun, Wei Hu, Dr. Peng Cui, Yan Wang and

Xianqiang Sun for the precious friendship. I am also thankful to Fuming Ying,

Dr. Hongmei Zhong, Dr. Jiayuan Qi, Dr. Haiyan Qin, Dr. Hao Ren, Dr. Qiu

Fang, Sathya and all colleagues in our department for the happy time we shared

together.

Special thanks are sent to Dr. Wei-Wei Zhang, Dr. Weijie Hua, Guangjun

Tian, Dr. Sai Duan, Dr. Yuejie Ai, Dr. Qiong Zhang, Yongfei Ji, Li Li and Xinrui

Cao, for their precious suggestions to improve this thesis.

Finally, I would like to express my special gratitude to my parents. Owing

to their endless love and strong support every time and every where, I have the

courage to face upto any difficulty and walk steadily forward.

Xing Chen

Fall 2011, Stockholm

Abbreviations

BO Born-Oppenheimier

CASSCF complete active space self-consistent-field

CSF configuration state function

CI conical intersection

DFT density functional theory

DFT-RU restricted-unrestricted density functional theory

EPR electron paramagnetic resonance

ESR electron spin resonance

FC Franck-Condon

GGA generalized gradient approximations

HF Hartree-Fock

HK Hohenberg-Kohn

HFCC hyperfine coupling constant

HOMO highest occupied molecular orbital

IC internal conversion

ISC intersystem crossing

KS Kohn-Sham

LUMO lowest unoccupied molecular orbital

LCAO linear combination of atomic orbitals

L(S)DA Local (spin) density approximations

LR-TDDFT linear response TDDFT

LCM linear coupling model

MCSCF Multi-configurational self-consistent field

MRCI multireference configuration interaction

xi

xii Abbreviations

NMR nuclear magnetic resonance

PES potential energy surface

RASSCF restricted active space self-consistent-field

RG Runge-Gross

SCF self-consistent-field

SOC spin-orbit coupling

TDDFT Time-depend density functional theory

TS transition state

UKS unrestricted Kohn-Sham

VR vibrational relaxation

ZPVC zero-point vibrational correction

List of Figures

2.1 The partition of orbital space in MCSCF approach . . . . . . . . . . . . 9

3.1 Sketch of magnetic resonance spectrometer . . . . . . . . . . . . . . . . 20

3.2 Splitted energy level in magnetic field for a molecule with S = 12 and

I = 12 , where E1 = µBg

isoB + 12A and E2 = µBg

isoB − 12A. . . . . . . . . 23

3.3 Schematic illustration of (a) Franck-Condon principle involving S1 withdominating transition (0-2), in an ideal condition of S0 with the same nor-mal modes as S1, which leads to (a) the mirror relationship of absorptionand emission spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1 Jablonski diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Schematic view of the photochemical reaction pathways. . . . . . . . . . 35

4.3 Schematic presentation of the reaction pathways along reaction coordi-nates on adiabatic PESs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 Schematic description of (a) transition state and (b) conical intersection. 39

4.5 Topological surface intersections (a)(n-2)-dimensional conical intersec-tion (b)(n-1)-dimensional surface crossing. . . . . . . . . . . . . . . . . . 40

5.1 The five allylic radicals. Selected from Paper 1, reprinted with permissionfrom The Royal Society of Chemistry. . . . . . . . . . . . . . . . . . . . . 44

5.2 Organic π radicals for which isotropic carbon and hydrogen HFCCs hasbeen computed with zero-point vibrational corrections. . . . . . . . . . . 45

5.3 Dependence between hydrogen and carbon HFCCs in organic π radicalanions: (a) computed at equilibrium geometry (b) with zero-point vibra-tional corrections added. Greenand blue denotes C2 positions for which inp-benzoquinone and fulvalene anions McConnell relation predict differentsign for the carbon HFCC. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.4 Intermediates and products derived from α-santonin reactions. . . . . . 46

5.5 Vibronically resolved spectra of photosantonic acid simulated at the CASS-CF and TDDFT levels in comparison with the experimental spectra. . . 47

5.6 Photochemical reaction pathways of α-santonin and lumisantonin sug-gested by experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.7 The Franck-Condon region of (a)α-santonin (b)lumisantonin. . . . . . . 50

5.8 Oxidation reaction of the computational models from the reactants tothe fist intermediates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.9 Schematic view of molecular orbtials and the electron occupancies of min-imum energy crossing points (a) (S/T)SC-I, (b) (S/T)SC-II and (c)(S/T)SC-III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

xiii

Contents

Abbreviations xi

List of Figures xiii

1 Introduction 1

2 Basic computational methods 3

2.1 Ab Initio theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Configuration interaction . . . . . . . . . . . . . . . . . . . 6

2.1.3 Multiconfiguration self-consistent field method . . . . . . . 7

2.1.4 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . 10

2.2 First principles theory . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Density functional theory . . . . . . . . . . . . . . . . . . 10

2.3 Response theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Time-dependent density functional theory . . . . . . . . . 17

3 Molecular spectroscopy 19

3.1 Magnetic resonance spectroscopy . . . . . . . . . . . . . . . . . . 19

3.1.1 Spin Hamiltonian approach . . . . . . . . . . . . . . . . . 20

3.1.2 EPR spin Hamiltonian . . . . . . . . . . . . . . . . . . . . 21

3.1.3 Hyperfine coupling tensor . . . . . . . . . . . . . . . . . . 22

3.2 Vibrationally resolved electronic spectroscopy . . . . . . . . . . . 27

3.2.1 Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . . . 28

3.2.2 Franck-Condon principle . . . . . . . . . . . . . . . . . . . 29

xv

xvi CONTENTS

4 Photochemical processes 33

4.1 Potential energy surface . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Surface intersection . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3 Nonadiabatic transition state theory . . . . . . . . . . . . . . . . 40

5 Summary of included papers 43

5.1 Molecular spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1.1 Hydrogen hyperfine coupling tensor in organic radicals, Pa-

pers 1-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1.2 Vibronically resolved spectroscopy of α-santonin derivatives,

Paper 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Nonadiabatic reactions . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2.1 α-Santonin photo-induced reactions, Papers 4-5 . . . . . . 48

5.2.2 Reaction of Cofactor-Free Enzyme with Triplet Oxygen Molecule,

Paper 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

References 55

Chapter 1

Introduction

Organic molecules with peculiar magnetic or optical properties have attracted

much scientific attention in many fields, including medicine, industry and mate-

rial science. Spectroscopy is a measurement of the system in response to an ex-

ternal perturbation of, for example, light or magnetic field. Molecular structures

can be identified from the magnetic resonance spectroscopies and/or the visible,

ultraviolet, and the fluorescence spectroscopies. Magnetic resonance phenome-

na illustrate an intrinsic property of magnetic molecules in an external magnetic

field. They can be divided into two types depending on the particles that interact

with the magnetic field: electron paramagnetic resonance (EPR), also referred to

as electron spin resonance (ESR), and nuclear magnetic resonance (NMR). EPR

spectroscopy is a technology adopted to describe the spatial distribution of para-

magnetic species [1]. The underlying physics is that the unpaired electrons with

spin magnetic moments can absorb or emit electromagnetic radiation depending

on the strength of the external magnetic field. Nowadays, with the development

of EPR technology, it has been widely used to monitor the presence of radicals in

chemical reactions, as well as to investigate biological and medical molecules with

non-vanishing total electron spins and magnetic materials [2].

It’s a challenging and complicated task to extract useful and significant in-

formation from EPR spectroscopy in order to make assignments of molecular

electronic and geometrical structures. A way to understand the experimental ob-

servations is by theoretical calculation of the effective spin Hamiltonian which

gives a relation between experimental data and predictions of quantum mechan-

ics. Nowadays, ab initio calculations play an important role in the prediction of

EPR parameters. These are calculations with high accuracy but which have the

disadvantage of high computational cost which limits their applicability to small-

1

2 1 Introduction

sized molecules. With the development of the density functional theory (DFT),

theoretical calculations of the EPR parameters can shed light on moderate- and

large-sized molecules with reasonable computational cost and accuracy.

Electronic absorption/emission spectroscopy is an important tool in analyti-

cal chemistry, where matter-radiation interaction between a sample and photons

is measured. These spectra contain important information about the excited-

state properties of molecules. The position and shape of a band in an electronic

spectrum are determined by the nature of the molecular electronic structure. Ac-

cordingly, the comparison between the simulated and the experimental spectra

provides accurate assignment of the transitions, the molecular electronic struc-

ture and the excited-state properties.

As stated by the Grotthuss-Draper law, a photochemical reaction takes place

when a photon is absorbed by a molecule. Photochemistry is tied up with our

daily life, for instance, photosynthesis takes place every moment in organisms. In

fact, the first organic molecule to be discovered with a photo-induced reactivity

was α-santonin, in the 1830s. Since then, many researchers have been attracted to

study of photoreaction and the photochemical techniques have been widely used

in industrial synthesis and other fields [3]. The photochemical reaction, chemical

reaction induced by light, is quite different from the conventional thermal reactions

on the ground-state potential energy surface (PES). As usually more than two

PESs are involved, investigations of the mechanism for a photochemical reaction

is challenging both in computations and experiments.

This thesis is devoted to the quantum mechanical simulations of two kinds

of spectra, the EPR and the electronic spectra, and two types of reactions, the

photochemical and spin-forbidden reactions. Special attention is payed to one of

the EPR parameters, the isotropic hyperfine coupling constant (HFCC). In the

following chapters, the basic computational approaches employed in the investi-

gations is first introduced in Chapter 2. Then, the essential concepts and basic

principles in molecular spectroscopy are represented in Chapter 3. Next, Chap-

ter 4 discusses the photochemical process and spin-forbidden reactions. Finally, a

summary of included papers is provided in Chapter 5.

Chapter 2

Basic computational methods

Ab initio and density functional theory are the basic methodologies to investi-

gate electronic structures of molecules. In molecular quantum mechanics, the

calculations of the energy and the wavefunction in a specific electronic state of a

medium-sized molecule are feasible under the Born-Oppenheimer (BO) approxi-

mation. Owing to the large ratio of nuclear mass to electronic mass, the nuclear

coordinates are regarded as constants in the potential energy term, and the to-

tal wavefunctions are successfully divided into electronic and nuclear (vibrational,

rotational) parts. Accordingly, the molecular wavefunction is represented by [4],

Ψt(r,R) = ΨN(R)Ψe(r,R). (2.1)

Although the BO approximation introduces some errors, the errors still can be

acceptable for the many-electron systems [5]. The basic computational methods

summarized below are in the framework of the BO approximation.

An ab initio method is a parameter-free method to solve Schrodinger equa-

tion [6–8]. The Hartree-Fock (HF) approximation is the simplest of ab initio theo-

ries. It is based on the approximation that each electron moves separately in the

average potential field arising from the remaining electrons, which means that the

electron-electron interaction is approximately described by a mean-field approach.

Thereby the many-body Schrodinger equation turns into an effective one-electron

equation, and the Coulomb interaction is treated in an averaged way. A con-

sequence of this approximation is that the probability of two close electrons is

overestimated, and the strength of coulomb interaction is enhanced. In quantum

mechanics, the correlation energy (Ecor) is defined by [9]

Ecor = E0 − EHF, (2.2)

3

4 2 Basic computational methods

where E0 is the exact lowest electronic energy in a given basis set in the BO

approximation. The Ecor consists of two parts: dynamical (short-ranged) orig-

inating from the instantaneous repulsion of electrons and static (long-ranged)

arising from the near-degeneracy of different electronic configurations in energy.

The Ecor gives much more contributions to some specific cases, such as molec-

ular dissociation reactions and excited state calculations, which are poorly de-

scribed by HF method. The post-Hartree-Fock methods use the HF calculation

as starting point and further improve the HF result by taking account of Ecor,

but are computationally more expensive. The post-HF methods include config-

uration interaction (CI) [10,11], coupled cluster (CC) [12–14], Møller-Plesset pertur-

bation theory (MP2 [15], MP3, MP4 [16], etc.), quadratic configuration interaction

(QCI [17]) and quantum chemistry composite methods (G2 [18], G3 [19], CBS, T1,

etc). Other post-HF methods include multiconfigurational self-consistent field

(MCSCF) [8,20,21], and as a special case, the complete active space SCF (CASS-

CF) [22,23]. The multireference methods taking Ecor into account, are extensions

of single reference methods (HF or CI methods), which take multi-configurational

wave functions as starting points for higher order appoximations – multireference

perturbation theory (MRPT, CASPT2 [24,25]) as well as multireference configu-

ration interaction (MRCI) [26] belong to these types. Density functional theory

(DFT) differing from ab initio methods is an alternative methodology in quantum

mechanics, and also considers the Ecor.

In our previous works summarized in this thesis, we used the popular compu-

tational methods (CASSCF and CASPT2) in the excited-state studies of the pho-

tolysis mechanisms of santonin and lumisantonin. Time-dependent DFT (DFT)

was also used in the excited-state optimizations in our studies.

For the evaluation of isotropic HFCCs, we used the restricted-unrestricted

DFT (DFT-RU) approach [27,28] to avoid the spin-contamination problems intro-

duced by unrestricted Kohn-Sham (UKS) formalism. The details in DFT-RU is

given in the following section.

2.1 Ab Initio theory 5

2.1 Ab Initio theory

2.1.1 Hartree-Fock

As mentioned above, in the HF approximation many-body wavefunctions are rep-

resented by an antisymmetrized product, the Slater determinant given by [29],

Ψ(χ1, χ2, . . . , χN) =1√N !

∣∣∣∣∣∣∣∣∣∣ϕ1(χ1) ϕ2(χ1) · · · ϕN(χ1)

ϕ1(χ2) ϕ2(χ2) · · · ϕN(χ2)...

.... . .

...

ϕ1(χN) ϕ2(χN) · · · ϕN(χN)

∣∣∣∣∣∣∣∣∣∣, (2.3)

where ϕi is a set of orthonormal orbitals - single electron wave functions. The total

electronic Hamiltonian of a molecule with M nuclei N electrons at fixed nuclear

geometry is written as [6],

He = −1

2

N∑i=1

∇2i −

M∑a=1

N∑i=1

Za

ra+∑i<j

1

rij. (2.4)

The HF energy is given by,

EHF =− 1

2

N∑i=1

⟨ϕi|∇2|ϕi⟩ −M∑a=1

N∑i=1

⟨ϕi|Za

ra|ϕi⟩

+∑i<j

⟨ϕi(1)ϕj(2)|1

r12|ϕi(1)ϕj(2)⟩ −

∑i<j

⟨ϕi(1)ϕj(2)|1

r12|ϕi(2)ϕj(1)⟩

=N∑i=1

Hcoreii +

N∑i=1

N∑j>i

(Jij −Kij),

(2.5)

Hcoreii ≡ ⟨ϕi|H|ϕi⟩, (2.6)

where H = −12∇2

i +∑M

a=1Za

riais one-electron operator, Jij and Kij are Coulomb

and exchange integrals, respectively.

To solve the HF problem is to find the single Slater determinant which yields

the lowest energy by means of the variational principle. This leads to the canonical

HF equations [21,30,31]

f(1)ϕi(1) = εiϕi(1), (2.7)


where εi represents the orbital energy, and Fock operator is defined by

fi(1) ≡ H(1) +∑j =i

(Jj(1)−Kj(1)) , (2.8)

where Jj and Kj are Coulomb operator and exchange operator, respectively. They

are defined by [32],

Jjφ(1) =

∫ϕ∗j(2)ϕj(2)

r12dτ2φ(1),

Kjφ(1) =

∫ϕ∗j(2)φ(2)

r12dτ2ϕj(1).

(2.9)

The Coulomb operator Jj stands for the potential energy of the interaction be-

tween electron 1 and the remaining electrons, and the exchange operatorKj arising

from the antisymmetric property of wavefunction has no counterpart in classical

physics. Finally, the total energy is calculated by [6],

EHF =∑i

εii −1

2

∑ij

(Jij −Kij), (2.10)

where,

Jij =⟨ϕi(1)ϕj(2)|r−112 |ϕi(1)ϕj(2)⟩,

Kij =⟨ϕi(1)ϕj(2)|r−112 |ϕi(2)ϕj(1)⟩.

(2.11)

2.1.2 Configuration interaction

Configuration interaction (CI) is the oldest approach which accounts for Ecor. The

trial wavefunction in the HF method is represented by a single Slater determinant,

whereas the total CI wavefunction is written as a linear combination of the HF de-

terminant and Slater determinants corresponding to excited-state configurations,

and the expansion coefficients are determined to give the lowest total energy [33].

The CI wavefunction is given by [6,8],

|Ψ⟩ = c0|Ψ0⟩+∑ai

cia|Ψia⟩+

∑a<bi<j

cijab|Ψijab⟩+

∑a<b<ci<j<k

cijkabc|Ψijkabc⟩+

∑a<b<c<di<j<k<l

cijklabcd|Ψijklabcd⟩+· · · ,

(2.12)

where the coefficients may be chosen to normalize the total wave function, or to

satisfy c0 = 1 in the case of intermediate normalization ⟨Ψ0|Ψ⟩ = 1. abcd etc.

stand for the occupied HF orbitals, ijkl etc. represent virtual orbitals. |Ψia⟩ is


singly excited determinant, which differs from |Ψ0⟩ by orbital ϕa replacing ϕi.

|Ψijab⟩ is doubly excited determinant, which means two electrons are promoted

from ab to ij, and so on. The combination coefficients, c0,cia, c

ijab,c

ijkabc,c

ijklabcd · · · ,

represent the contribution of each configuration to the total CI wavefunction.

The CI wave function consisting of all possible configurations is named full CI,

which gives the exact nonrelativistic ground state energy of the system within the

BO approximation for a given basis set. According to Brillouin’s theorem [8] and

Slater-Condon rules [29,34], the total energy is given by [6],

E = ⟨Ψ0|He|Ψ⟩

= ⟨Ψ0|He|Ψ0⟩+∑a<bi<j

cijab⟨Ψ0|He|Ψijab⟩

= EHF +∑a<bi<j

cijab⟨Ψ0|He|Ψijab⟩.

(2.13)

From the definition of Ecor, it can be calculated by,

Ecor = E − EHF =∑a<bi<j

cijab⟨Ψ0|He|Ψijab⟩. (2.14)

Evidently, the main contribution to Ecor is solely arising from the doubly excited

configuration. The determination of Ec requires the values of all configuration

coefficients, because cijab is coupled with all the other coefficients. In other word-

s, the eigenvector of full CI matrix should be known. It is a time consuming

method, accordingly confined to the systems with few electrons using very small

basis sets. There are approximate methods taking account of the most important

configurations in the electron wavefunctions, such as CISD [35], which means the

wavefunction is truncated after the doubly excited configurations. The singly ex-

cited CI (CIS) - truncation of the series after singly excited configurations [36] is

the simplest of CI methods to compute excited states.

2.1.3 Multiconfiguration self-consistent field method

In some cases, single configuration and single reference methods poorly describe

the electronic states of system with near-degenerate configuration state function-

s (CSFs) [36]. One popular approach employed to research the excited states is

the multiconfiguration self-consistent field method (MCSCF) [8,20,21]. The MCSCF


wavefunction is given by a linear combination of CSFs, as shown in (2.15) [6,8],

ΨMCSCF =∑i

CiΦi, (2.15)

where Φi is a CSF, and it can be expressed as a determinant constructed by

molecular orbitals, see (2.16)

Φi =1√N !Det

[∏j⊂i

ϕj

]. (2.16)

and the molecular orbital is a linear combination of atomic basis functions,

ϕj =∑ν

Cνjχν . (2.17)

In the MCSCF method, the total energy is a functional of expansion coef-

ficients, Ci and orbital coefficients Cνj, which are simultaneously optimized to

obtain the lowest energy.

The MCSCF approach provides a valid tool to give an accurate static corre-

lation energy considering a relatively small number of CSFs. Moreover, MCSCF

is free of the spin contamination problem for the open-shell systems. An essential

task in an MCSCF calculation is the choice of configurations which is dependent

on the nature of studied problem, i.e. it is not a ‘black box’ method. Usually,

the configurations originate from the lowest electronic excitations the orbitals of

which define an active space. A special case is called CASSCF [37] which consid-

ers all possible configurations within the active space. In the CASSCF approach,

the molecular orbital space is divided into three subspaces: inactive, active and

external orbitals. All the orbitals are doubly occupied in the inactive space, the re-

maining electrons occupy in the active space, and the external orbitals are virtual

orbitals in all configurations. Once the electrons distributed into the active space

is defined, the number of CSFs regardless of molecular symmetry is determined

by the Weyl-Robinson equation [38],

K(n,N, S) =2S + 1

n+ 1

(n+ 1

12N − S

)(n+ 1

12N + S + 1

), (2.18)

where n is the number of electrons distributed into the active space, m is the

number of active orbitals where S is the total spin of electronic state. The key

problem in CASSCF calculations is the choice of active space. The selected rules


Frozen

Inactive

0 ~ n excitations out (RAS1)

Complete active space (RAS2)

0 ~ n excitations in (RAS3)

Virtual

Figure 2.1 The partition of orbital space in MCSCF approach

can not easily be generalized because of the specific problems in different systems.

The rule of thumb is that all orbitals related to the excited states we are con-

cerned about or are involved in the chemical reactions should belong to the active

space. For the small conjugated organics, the active space usually includes π-type

orbitals in order to obtain the ππ∗ state. In some cases, the nπ∗ state is quite

significant, therefore, lone pairs should be taken into account. An extension to

the CASSCF approach is restricted active space SCF (RASSCF) method [39]. The

active space is further divided into three subspaces: RAS1, RAS2 and RAS3. In

RAS1, the orbitals are doubly occupied, but a limited number of electrons allowed

to be promoted from the RAS1 to RAS2 or RAS3 subspace. A given number of

electrons are distributed into RAS2 subspace, where the CSFs generated from

full CI method. In RAS3, only limited number of electrons is allowed, and the

excited electrons either come from the RAS1 or RAS2 subspace. If the orbitals

in RAS1 are fully occupied and RAS3 is totally empty, the RASSCF is equal to

CASSCF. The orbital partitioning is schematically illustrated in Fig. 2.1. The

advantage of RASSCF method is to enlarge the active space without exploding

the CI expansion. Therefore, it has the potential to be applied to systems where

the CASSCF method is not possible to use.


2.1.4 Perturbation theory

As mentioned previously, in CASSCF the CSFs are composed of the near-degenerate

configurations, so that static correlation is sufficiently taken into account. How-

ever, a lack of dynamic correlation gives inaccurate energies at equilibrium ge-

ometries of excited states and surface intersections. An efficient way to improve

the CASSCF approach with dynamic correlation is based on perturbation theory,

namely CASPT2 [25]. The spirit of this method is to calculate the second-order

energy with CASSCF wavefunction as zeroth-order approximation. The zeroth-

order Hamiltonian is defined by one-electron Hamiltonian in a convenient way.

However, the CASPT2 method is inappropriate when the electronic states with

same symmetry are near-degenerate. As an extension of CASPT2, multi-state

perturbative methods developed by Nakano and Finley et al [40,41] can handle such

problem well. Nowadays, CASPT2 is employed to explore the stationary points

or surface intersections in potential energy surfaces (PES). In our previous work

included in this thesis, the CASPT2 method was used to evaluate the energies of

the CASSCF-optimized key points along the reaction pathways on the PESs.

2.2 First principles theory

2.2.1 Density functional theory

Density Functional Theory (DFT) stems from the work by Thomas and Fermi in

the early 1920s [42,43], and has made a significant impact in quantum chemistry.

Because of moderate computational cost and high accuracy compared with ab i-

nitio methods, it is widely used in many fields, such as computational chemistry,

solid state physics, and computational biology. It has grown to be one of the

brilliant and glamorous stars in the vast galaxy of numerous computational meth-

ods by now. The inherent advantages of efficient scaling with the size of system

and an accurate description of ground-state properties by using an appropriate

exchange-correlation functional, it is a popular tool for exploring many interesting

properties of large-sized systems.

Kohn-Sham approach

The corner stone of the Kohn-Sham (KS) approach is the Hohenberg-Kohn (HK)

theorem [44] proposed by Hohenberg and Kohn in 1964. They provided a solid proof

2.2 First principles theory 11

to verify that DFT is an exact theory for the description of electronic structure of

matter. The remarkable theorem is stated

There exists a variational principle in terms of the electron density which

determines the ground state energy and electron density. Further, the ground state

electron density determines the external potential, within an additive constant.

It indicates the ground state energy E is a functional of electron density

ρ(r) and external potential v, that is E = E[ρ(r), v], and a trial well-behaved

density corresponds to an energy which is higher or equal to the exact energy

(E0). Accordingly, the energy of ground state is given by [44,45],

E0[ρ] = F [ρ] +

∫vextρdr, (2.19)

where F [ρ] is the universal functional consisting of electron kinetic energy and

electron-electron repulsion energy. Using Lagrange’s method with undetermined

multipliers, there is

δ

(E(ρ)− µ(

∫ρdr−N)

)= 0. (2.20)

Eq. (2.20) is rewritten as an Euler equation,

δF [ρ]

δρ+ vext = µ. (2.21)

Here µ is a Lagrange multiplier, implying the change in chemical potential arising

from different electron densities, and N is the total number of electrons. However,

the HK theorem doesn’t point out how to calculate E from ρ because of the form

of F [ρ] unknown or how to find ρ.

Nowadays, the most popular methods of DFT are based on KS scheme [46].

It introduces a reference system with N noninteracting electrons moving in an

effective potential vs, the Hamiltonian is written as [46],

Hs = −1

2

∑i

∇2i +

∑i

vs(ri). (2.22)

The energy is given by

Es = Ts[ρ] +

∫vsρdr. (2.23)

Here Ts[ρ] is kinetic functional of reference system. Derived from Eq. (2.21) and

(2.23), there isδTs[ρ]

δρ+ vs = µs. (2.24)


For a real system, F [ρ] is expressed as,

F [ρ] = T [ρ] + Vee[ρ]

= Ts[ρ] + J [ρ] + (Vee[ρ]− J [ρ] + T [ρ]− Ts[ρ])= Ts[ρ] + J [ρ] + Exc[ρ],

(2.25)

where J [ρ] is coulomb repulsion represented as, 12

∫ρ(r1)ρ(r2)r

−112 dr1dr2, and Exc[ρ]

is exchange and correlation functionals. Eq. (2.21) is rewritten as,

δTs[ρ]

δρ+

∫ρ(r2)

r12dr2 + vxc + vext = µ. (2.26)

According to HK theorem, the same electron density yields the same external po-

tential. Compared Eq. (2.26) with Eq. (2.24), KS effective potential is represented

as,

vs =

∫ρ(r2)

r12dr2 + vxc + vext. (2.27)

The spirit of the KS method is to search for the density which minimizes

the energy for an approximate Exc[ρ][45,47]. Given Exc[ρ], the exchange-correlation

potential vxc is determined, which is used for the construction of the KS potential

vs. The solution of the KS equations gives the KS orbitals, which are used to con-

struct electron density, and the ground-state energy. An iterative method must

be applied because the construction of vs requires the electron density. A conve-

nient way to obtain the KS orbitals is by employment of the linear combination

of atomic orbitals (LCAO) method, in analogy to Hartree-Fock-Roothan theory.

Therefore, the KS orbitals are represented in terms of basis functions χµ,

|ϕi⟩ =M∑µ

χµcµi, (2.28)

where M is the number of basis functions. The KS equations are written as

f |ϕi⟩ = ϵi|ϕi⟩,

f = −1

2∇2 + vs.

(2.29)

Spin-restricted Kohn-Sham approach

In an paramagnetic molecules, i.e. open-shell system, the spin orbitals require

additional care, as the α and β densities are different. In an open-shell molecule,


the energy is rewritten as,

E[ρα, ρβ] =Ts[ρα, ρβ] + J [ρα, ρβ]

+ Exc[ρα, ρβ] +

∫(ρα, ρβ)vextdr,

(2.30)

where ρα and ρβ are α and β spin densities, respectively, which are defined by,

ρα =∑i

nαi|ϕαi |2,

ρβ =∑i

nβi|ϕβi |2.

(2.31)

nαi represents an occupation number in spin orbital ϕαi . And the Kohn-Sham

equations for α and β spin orbitals are given by,

fαϕαi =

ϵαinαi

ϕαi ,

fβϕβj =

ϵβjnβj

ϕβj .

(2.32)

where, i and j run over all the α and β orbitals, respectively. The operators are

defined,

fα =− 1

2∇2 + vαs = −1

2∇2 + vext +

∫ρα(r2) + ρβ(r2)

r12dr2

+δExc[ρα, ρβ]

δρα,

(2.33)

fβ =− 1

2∇2 + vβs = −1

2∇2 + vext +

∫ρα(r2) + ρβ(r2)

r12dr2

+δExc[ρα, ρβ]

δρβ,

(2.34)

By using the variational method, the minimization of energy is performed

with respect to α and β spin densities, respectively. There are two solutions in

this procedure differing by the variational constraints. One is the unrestricted KS

approach dealing with the α and β orbitals separately. In this method, two KS

matrixes are generated and diagonalized in each iteration for the exploration of

the lowest energy. The disadvantage of this method is the introduction of spin

contamination. The other is spin-restricted KS approach, which divides orbitals

into double-occupied and single-occupied types. This solution avoids the spin

contamination problem, but it leads to the off-diagonal Lagrangian multipliers.


There exist two methods to cope with this difficulty: one is handle the equations

of double-occupied and single-occupied orbitals individually, and the other is a

construction of ‘effective equation’ with off-diagonal Lagrangian multipliers. In

our previous works, we adopted the latter to treat the open-shell molecules to

calculate the magnetic properties. I give a brief introduction of spin-restricted KS

approach with the latter solution in this section.

In the spin-restricted approach, the KS equations are given by,

fdϕk =

Nd+Ns∑j

ϵkjϕj,

1

2fsϕl =

Nd+Ns∑j

ϵljϕj,

(2.35)

where Nd and Ns are the total number of doubly-occupied and singly-occupied

orbitals, respectively. k runs over the doubly occupied orbtials and l runs over the

singly occupied orbitals. The KS operators are represented as,

fd =−1

2∇2 + vds = −1

2∇2 + vext +

∫ρα(r2) + ρβ(r2)

r12dr2

+1

2

δExc[ρα, ρβ]

δρα+

1

2

δExc[ρα, ρβ]

δρβ,

(2.36)

fs =−1

2∇2 + vss = −

1

2∇2 + vext +

∫ρα(r2) + ρβ(r2)

r12dr2

+δExc[ρα, ρβ]

δρα,

(2.37)

where the α and β spin densities are written as

ρα =

Nd∑i=1

|ϕαi |2 +

Ns∑j=1

|ϕαj |2,

ρβ =

Nd∑i=1

|ϕβi |2.

(2.38)

Accordingly, the effective KS matrix is defined by,

H =

Fd Fds Fs

Fds Fs Fsv

Fs Fsv Fd

. (2.39)


The Fµν are defined by,

Fd =1

2(fα + fβ),

Fds = fβ,

Fsv = fα,

(2.40)

where fα and fβ stand for the matrix elements which are defined in Eq. (2.32). It

should be mentioned that the eigenvalues of the effective matrix have no physical

meaning, it is only a construction for minimizing the energy in spin-restricted

calculations.

Exchange-correlation functional

As the exact exchange-correlation functional is unknown, approximations are re-

quired, generally fittings of appropriately parameterized functionals in order to

describe the total energy within the framework of the KS approach. The exchange-

correlation functionals are mainly composed of the following types:

Local (spin) density approximations (L(S)DA) It is a coarse approxi-

mation to the real functional. The energy volume density is solely dependent on

the electron density at a given position [48],

Exc[ρα, ρβ] =

∫ρ(r)εxc(ρα(r), ρβ(r))dτ. (2.41)

where ρα and ρβ are α and β spin densities, respectively. εxc is the exchange-

correlation energy per particle in the uniform electron gas approximation.

Generalized gradient approximations (GGA) The density and the gra-

dient of density ∇ρ(r) are the variables in the exchange-correlation functional. It

has better performance in regions of the highly varying electron density [49,50].

Exc[ρα, ρβ] =

∫f(ρα(r), ρβ(r);∇ρα(r)∇ρβ(r))dτ. (2.42)

Hybrid functional This functional has in addition to GGA contributions,

a fraction of the Hartree-Fock exchange term [51], The most common hybrid func-

tional B3LYP [52] was first proposed by Becke, and is well suited for predictions of

energetics with in the KS framework,

Exc = ELDAxc +0.72(EB88

x −ELDAx )+0.81(ELY P

c −ELDAc )+0.2(EHF

x −ELDAx ) (2.43)


2.3 Response theory

The interaction between a molecule and an external electric field can be described

with response theory, i.e. a formalism for time-dependent perturbation theory

which has time-independent perturbation theory as a special case.

The molecular properties are computed as response functions [53]. Under a

time-dependent perturbation field, the Hamiltonian (H) can be written as a com-

bination of the unperturbed HamiltonianH0 and the time-dependent perturbation

V(t),H = H0 + V(t). (2.44)

The perturbation operator can be expressed by Fourier transform in the frequency

domain,

V(t) =∫ +∞

−∞Vωexp[(−iω + ε)t]dω. (2.45)

The perturbation field contains a positive infinitesimal ε, such that V(t) is zero at

t = −∞. Vω satisfies the hermiticity condition,

(Vω)† = V(−ω). (2.46)

The perturbed wavefunction can be rewritten as,

|0(t)⟩ =|0⟩+∫ +∞

−∞|0(ω1)

1 ⟩exp[(−iω + ε)t]dω1

+

∫ +∞

−∞

∫ +∞

−∞|0(ω1,ω2)

2 ⟩exp[(−i(ω1 + ω2) + 2ε)t]dω1dω2 + · · · .(2.47)

Here the second and third terms describe the wavefunction changes in linear and

quadratic response to a perturbation, respectively. The expectation value can be

derived at the time-dependent perturbation,

⟨0(t)|A|0(t)⟩ =⟨0|A|0⟩+∫ +∞

−∞⟨⟨A;Vω1⟩⟩ω1exp[(−iω + ε)t]dω1

+1

2

∫ +∞

−∞

∫ +∞

−∞⟨⟨A;Vω1 ,Vω2⟩⟩exp[(−i(ω1 + ω2) + 2ε)t]dω1dω2 + · · · .

(2.48)

Here, the first term on the right-hand of the equation is the expectation value of

the operator A without a perturbation. The second and third terms corresponds

to the linear and quadratic response functions, respectively.

2.3 Response theory 17

2.3.1 Time-dependent density functional theory

Time-dependent density functional theory (TDDFT) is based on the Runge-Gross

(RG) theorem [54], which states that the potential is a functional of the time-

dependent density. The time-dependent KS (TDKS) equations are written as,

i∂

∂tϕi(r, t) =

(−1

2∇2

i + vs(r, t)

)ϕi(r, t), (2.49)

with the density

ρ(r, t) =N∑i

|ϕi(r, t)|2. (2.50)

vs(r, t) is represented as,

vs(r, t) = vext(r, t) + vJ(r, t) + vxc(r, t), (2.51)

where vJ(r, t) is time-dependent Coulomb potential, written as vJ(r, t) =∫d3r′ ρ(r

′,t)|r−r′| .

vxc is exchange-correlation potential.

The most common method to solve the TDDFT equations is the linear re-

sponse TDDFT (LR-TDDFT); assuming a weak perturbing field, perturbation

theory is applied. vxc depends on the change of densities (ρ(r, t) = ρgs(r, t) +

δρ(r, t)) [55],

vxc[ρgs + δρ](r, t) = vxc[ρgs](r) +

∫dt′∫d3r′fxc[gs](r, r

′, t− t′)δρ(r′, t′) (2.52)

Here an important definition of fxc kernel is introduced[56]

fxc[ρgs](r, r′, t− t′) = δvxc(r, t)

δρ(r′, t′)|ρ=ρgs . (2.53)

As a functional of the ground state density solely, it simplifies the full time-

dependent exchange-correlation potential. Here, we give another definition of

point-wise susceptibility χ[ρgs](r, r′, t− t′), in order to show the importance of fxc,

δρ(r, t) =

∫dt′∫d3r′χ[ρgs](r, r

′, t− t′)δvext(r′, t′), (2.54)

where χ is the linear density-density response functional of the ground-state den-

sity. It tells us how the density changes at position r and later time t in response

to a tiny change in the external potential field at position r′. By analogy to


ground-state KS system, it also has χKS and the density response is given by,

δρ(r, t) =

∫dt′∫d3r′χKS[ρgs](r, r

′, t− t′)δvs(r′, t′). (2.55)

Accordingly, derived from Eq.(2.54) as well as using Fourier transform in frequency

domain, χ of the interacting system can be described as [55–58],

χ(r, r′, ω) =χKS(r, r′, ω)

+

∫d3r1

∫d3r2χKS(r, r1, ω){

1

|r1 − r2|+ fxc(r1, r2, ω)}χ(r2, r′, ω).

(2.56)

The equation carries information about the electronic excitations in TDDFT. As

ω is equal to a transition frequency, χ has a pole. By analogy, χKS has a series of

poles, the one-particle excitation in KS system [55],

χKS(r, r′, ω) = 2 lim

η→0+

∑q

(ξq(r)ξ

∗q (r

′)

ω − ωq + iη−

ξ∗q (r)ξq(r′)

ω + ωq − iη

), (2.57)

where, ξq(r) = φ∗i (r)φa(r). When the electron is promoted from the occupied KS

orbital φi to an unoccupied KS orbital φq, it is represented as,

ωq = εq − εi, (2.58)

where ε is the energy of KS orbital. As frequency equals ω, the excitation en-

ergy corresponds to the pole of the linear susceptibility χKS for the ground-state

potential.

Chapter 3

Molecular spectroscopy

With the development of accurate and efficient computational methods in recent

decades, theoretical investigations in molecular spectroscopy have become more

feasible. Various spectroscopies are widely used in many research fields, such

as clinical inspection, atmospheric monitoring and photochemical synthesis. In

this work we focus on magnetic resonance spectroscopy and vibrationally resolved

electronic spectroscopy.

3.1 Magnetic resonance spectroscopy

Magnetic resonance spectroscopy measures the irradiation absorption in response

to the strength of external magnetic field. The magnetic resonance phenomenon

includes two types: nuclear magnetic resonance and electron paramagnetic reso-

nance, referring to nuclear spin and electronic spin, respectively, which interact

with the magnetic field [59]. EPR spectroscopy is adopted for studying systems

with one or more unpaired electrons, which are capable of absorbing microwave

radiation in the presence of an external static magnetic field. The first experiment

on magnetic resonance can be traced back to 1945 [60], and the first report about

hyperfine structure of EPR spectrum was in 1949 [61]. However, the progress of the

EPR technique was not significant until the 1980s, the golden age of EPR devel-

opment. Then commercial pulsed spectrometers were developed and employed in

large scale and high-field EPR spectroscopy developed rapidly [62]. It is worthy to

mention that the first high-frequency pulsed EPR, produced by J. Schmidt et al.,

came out in 1988 [63]. The advanced EPR spectrometers provide high quality spec-

tra providing accurate analysis of experimental data. Nowadays, EPR has been

widely used in many fields to identify the structure of paramagnetic molecules.

19

20 3 Molecular spectroscopy

A magnetic resonance spectrometer is schematically illustrated in Fig. 3.1. An

BB

Signal

Spectroscopy

Sample

Magnet Magnet

Microwave generator Detetor Amplifier

Figure 3.1 Sketch of magnetic resonance spectrometer

electronic spin magnetic moment aligns itself parallel or antiparallel along the ex-

ternal magnetic field, leading to a split of the original energy level. The unpaired

electron undergoes a transition between the levels by absorption or emission of a

photon with a frequency which matches the energy gap between the split level-

s [64]. The most common photon frequency used in the EPR spectrometer is in the

range of 9000-10000 MHZ, with a magnetic field of about 3500 G, and a relaxation

with characteristic time scale from 10−4 to 10−8 s. In the following subsections,

I give an outline of the general theory of the spin Hamiltonian approach and its

applications in EPR.

3.1.1 Spin Hamiltonian approach

The spin Hamiltonian is like a bridge which connects the magnetic resonance

phenomenon calculated by using theoretical methods with experimental spec-

troscopy [1,59]. A reasonable assignment of the observed molecular electronic struc-

ture is derived from a comparison of the computed magnetic resonance parameters,

such as the g-tensor and the hyperfine coupling constants, with the experimental

data.

The spin Hamiltonian can be constructed in two ways, one roots in the exper-

imental data, namely the effective spin Hamiltonian and the other is true quantum

mechanical spin Hamiltonian, originating from relativistic quantum mechanics in

description of the particle mechanics and electromagnetic interactions. The effec-

tive spin Hamiltonian contains the parametric matrices fitted from experimental

3.1 Magnetic resonance spectroscopy 21

spectroscopy, while the true spin Hamiltonian is composed of differential opera-

tors coming from relativistic quantum mechanics in expression of physical effects

underlying the magnetic resonance spectra [59]. Taking advantage of both method-

ologies is significant to give an accurate estimate of the quantitative influence of

the magnetic field compared with the experimental observations.

The effective spin Hamiltonian is defined as a Hermitian operator in terms of

spin operators and parameters [1,59,65]. Generally, the effective spin Hamiltonian is

given by,

HS = βBS · ←→g e · B +∑N

S ·←→A N · IN + S ·

←→D · S

−∑N

βNB · ←→g N · (1− σN)IN +N∑i<j

Ii · (←→G ij +

←→K ij) · Ij.

(3.1)

Here βB and βN stand for Bohr and nuclear magnetons, respectively. ←→g e and←→g N represent the electronic and nuclear g-tensors, respectively.

←→A , the A-tensor,

predicts the hyperfine interaction arising from the electronic spin and the magnetic

field generated from the magnetic nuclei, and←→D and σN are called zero field

splitting tensor and the nth nuclear shielding tensor, respectively. The first and

forth terms in Eq. (3.1) represent the Zeeman effect on the resonance energy for

electronic and nuclear spin states, respectively. The second term is related to the

hyperfine structure in EPR. The third term refers to the spin-spin interactions

arising from the unpaired electrons. The last term describes the nuclear spin-spin

interaction including the classical (←→G ) and indirect (

←→K ) nuclear dipolar spin-spin

coupling tensors.

3.1.2 EPR spin Hamiltonian

The EPR phenomenon is the response of a molecule with non-vanishing electronic

spin angular momentum to an external static magnetic field and a radiation field

and is recorded by the EPR spectroscopy. The Zeeman effect removes the original-

ly degenerate levels of the electron spin states, and the transition energy between

the two splitted levels is identified. Although a molecule studied by EPR may

have the non-vanishing nuclear magnetic moments, the nuclear magnetic dipole

is only two parts per thousand in magnitude of the electronic magnetic dipole;

therefore, the chemical shifts and the nuclear spin-spin interactions do not appear

on the same energy scale as the electronic interactions. Consequently, the terms

related to nuclear magnetic dipole interactions in Eq. (3.1) can be discarded in the


construction of EPR spin Hamiltonian. Accordingly, the basic and conventional

effective spin Hamiltonian is defined as,

HS(EPR) = βBS · ←→g e · B +∑N

S ·←→A N · IN + S ·

←→D · S. (3.2)

The first term is the electronic Zeeman effect governing the position of the peaks

in the EPR spectra, the second term corresponds to hyperfine splitting term o-

riginating from the interaction of electronic spin with nuclear spin moment. The

third term stands for the high spin paramagnetism (S > 12) arising from the

dipole-dipole field-independent interaction between the electronic spins, namely

the zero-field splitting.

Fig. 3.2 illustrates an isotropic energy level split of a paramagnetic molecule

with one magnetic nucleus and one unpaired electron in the presence of an external

magnetic field of strength B0. Without the perturbation from the external static

magnetic field, all the spin states are degenerate. As the strength of external

magnetic field increases, the degenerate spin states are split into two sublevels

regardless of the coupling between the electronic and nuclear spins. Taking the

hyperfine coupling effects into account, each energy level is further split into two

sublevels. The electronic transition between different energy levels is required to

satisfy a selection rule concerned with a single excitation process in EPR, where

∆mS = ±1 and ∆mI = 0. The transition strengths are equal, which is reflected in

the EPR spectrum by two splitted peaks with same intensity, and their distance

is the isotropic hyperfine coupling constant.

3.1.3 Hyperfine coupling tensor

As mentioned above, the hyperfine coupling tensor, which represents the inter-

action between the unpaired electronic spin and the nuclear magnetic moment

gives rise to a hyperfine structure in EPR spectroscopy. Essential information

about the chemical environment surrounding a given nucleus and the spin density

distribution can be obtained from the hyperfine coupling tensors [66,67].

Ab initio studies on the hyperfine coupling interaction can be traced back to

1960s [68]. The estimation derived from Hartree-Fock theory greatly deviates from

experiment due to lack of electron correlation. Accordingly, the post-Hartree-Fock

approaches with electron correlation give the significantly improved results. How-

ever, correlated ab initio calculations of high-accuracy are expensive and confined

to light main group systems and small-sized organic radicals. The development


Ene

rgy

ms =+

12+

12+ 1

2+

12+

12

-

12

- 12

-

12

-

,

,

,

,

ms = -

Β0 = 0, Α = 0 Β0 > 0, Α = 0 Β0 > 0, Α > 0

12

12

E1 E2

Figure 3.2 Splitted energy level in magnetic field for a molecule with S = 12 and

I = 12 , where E1 = µBg

isoB + 12A and E2 = µBg

isoB − 12A.

of density functional theory (DFT) has enabled hyperfine coupling tensor calcula-

tions for large-sized systems as well [69–71]. By including electron correlation with

moderate computational cost, DFT is a good alternative for the prediction of hy-

perfine coupling tensors. The unrestricted Kohn-Sham formalism gives rise to spin

contamination leading to an unreliable results [72,73]. However, density function-

al restricted-unrestricted approach avoids this in magnetic property calculations.

The isotropic hyperfine coupling constant calculations displayed in this thesis are

carried out by DFT-RU methodology [27,28,74].

The spin Hamiltonian representing the hyperfine coupling interaction between

the electronic and nuclear magnetic moments is given by,

HHFC =∑N

S ·←→A N · IN . (3.3)


The hyperfine tensor←→A N consists of two contributions from the isotropic and

anisotropic (dipolar) interactions, differing by their physical origins and how they

are observed in experiment [59]. The former contribution is equal to the average val-

ue of non-classic Fermi contact interaction, measured in the gas phase or solution

by experimental means. The anisotropic contribution arises from the magnetic

dipolar interaction between the nuclear and electronic spins, and it should be con-

sidered in ordered samples, meaning that the molecules are oriented under the

external magnetic field. The isotropic interaction term is given by [75],

AisoN =

4

3~gegNβeβN⟨SZ⟩−1⟨Ψ|δ(r)|Ψ⟩. (3.4)

Here ge represents the g-value (2.002 319 31) of the free electron and gN the nuclear

g-value, which are the fundamental parameters responsible for the Zeeman effect.

⟨SZ⟩ is the expectation value of total electron spin along z-axis. ⟨Ψ|δ(r)|Ψ⟩ isthe spin density at a given nucleus. Generally, the electron spin density is highly

influenced by s orbital of nucleus. Consequently, the basis sets chosen for the Aiso

calculations should be good at describing a flexible inner core with a series of tight

s functions. The classical electron-nuclear dipole-dipole interaction term is given

by [75],

AdipN,ij = gegNβeβN⟨SZ⟩−1⟨Ψ|δijr

2 − 3rirjr5

|Ψ⟩, (3.5)

and the dipolar coupling tensor←→A dip

N is traceless and symmetric. The hyperfine

coupling tensor can be defined as the second-order derivative of the total electronic

energy with respect to electronic and nuclear spins given by,

←→A =

∂2E

∂S∂IN|S=0,IN=0. (3.6)

Isotropic hyperfine coupling constant by the restricted-unrestricted DFT

approach

The isotropic HFCC, namely the Fermi contact interaction, represents the inter-

action between magnetic moments of the nuclear and electronic spins in a para-

magnetic molecule [76,77], and can be calculated by first order perturbation theory.

In the KS-DFT scheme, a triplet perturbation can be taken into account in dif-

ferent ways. The unrestricted KS (UKS) formalism [78–81] is commonly employed

to evaluate HFCCs. It has the advantage of a simple formalism, but introduces

spin contamination. A spin-unrestricted formalism is not feasible because doubly


occupied core orbitals give zero spin density on the nucleus. Fortunately, the DFT-

RU approach avoids the shortcomings and inherits the advantages of UKS [27,28,74].

This approach is derived from the open-shell spin-restricted KS method for the

ground state and unrestricted approach to treat the triplet properties by pertur-

bations [27,28,74]. The isotropic HFCC of the n-th nucleus within the framework of

DFT-RU approach is written as,

AisoN = ⟨0| ∂

2HFC

∂Sz∂INz|0⟩∣∣∣∣Sz=0,INz =0

+ ⟨⟨ ∂2HFC

∂Sz∂INz;H0⟩⟩0,0

∣∣∣∣Sz=0,INz =0

, (3.7)

where Sz and INz are the Cartesian z component of the effective electronic spin

and the nuclear spin of the n-th magnetic nucleus, respectively. HFC stands for

the FC operator, andH0 represents the zeroth-order KS Hamiltonian without per-

turbation. The Fermi contact interaction arising from spin-density contribution is

calculated from the first term in the right-hand side of Eq. (3.7), and the second

term is the spin polarized correction to the Fermi contact interaction in terms of a

triplet linear response function [27,28]. In this way, the isotropic HFCC is rigorously

divided into two types: the direct spin density and spin polarized contributions.

Zero-point vibrational correction to isotropic hyperfine coupling con-

stant

The zero-point vibrational corrections (ZPVC) usually make a significant contri-

bution to the HFCC of semi-rigid paramagnetic species with several small frequen-

cies in the ground state. There exists some approaches for investigating ZPVC

effects. They are mainly divided into two types: perturbation theory and varia-

tional methodology. The earlier attempts were to fit the higher potential energy

derivatives to a potential energy functional at different geometries [82–85], which

need a large number of suitable energy sampling points. Furthermore, there is no

universal solution to all fitting problems. Due to its high cost, the application of

this method is obviously confined to small-sized systems, usually with no more

than four atoms. The perturbation approach employs a Taylor expansion of the

potential energy around the equilibrium geometry or so-called effective geome-

try [86–89]. The former takes account of the harmonic and anharmonic vibrational

corrections simultaneously in a perturbative manner, which could cause the con-

vergence problem arising from the existent higher derivative terms for large-sized

molecules. In this thesis, I concentrate on the latter approach where the harmonic

correction is computed at the effective geometry, and the anharmonic correction


is derived from the difference between the property estimated at both the equi-

librium and the effective geometries [87,90–92]. The effective geometry optimization

satisfy the following condition:

V(1)eff,j +

1

4

N∑i=1

V(3)eff,iij

ωeff,i

= 0. (3.8)

Here i and j denote vibrational modes, V(1)eff,j and V

(3)eff,iij are the first- and third-

order derivatives of the potential with respect to the normal coordinates, respec-

tively. ωeff,i stands for the harmonic frequency at the effective geometry for normal

mode i, and N is the total number of vibrational modes. The molecular effective

geometry is derived from the equilibrium geometry {Reql,K} following a manner

{Reff,K} = {Reql,K} −1

4ω2K

∑L

V(3)eql,KLL

ωL

, (3.9)

where V(3)eql,KLL represents the semi-diagonal elements of the cubic force field at the

equilibrium geometry. Accordingly, the effective geometry is related to the cubic

force with 6K-11 molecular hessian gradient along the normal coordinates, where

K is the total number of atoms in the concerned molecule. And HFCC including

ZPVC is given by,

Aisovib = Aiso

eff +1

4

N∑i=1

A(2)isoeff,ii

ωeff,i

, (3.10)

where the first term on the right-hand side of the Eq. (3.10) is the HFCC obtained

at the effective geometry, the second term describes the harmonic vibrational

contribution, in the form of the second derivative with respect to the normal

mode i. The anharmonic vibrational contribution is derived from the effective

geometry. The ZVPC is given by,

AisoZPV = Aiso

eff − Aisoeql + Aiso

har = Aisoanh + Aiso

har, (3.11)

with

Aisohar = Aiso

vib − Aisoeff , (3.12)

Aisoanh = Aiso

eff − Aisoeql. (3.13)

3.2 Vibrationally resolved electronic spectroscopy 27

3.2 Vibrationally resolved electronic spectroscopy

Electronic spectroscopy measures the transition energy between different elec-

tronic states by irradiation of ultraviolet (UV) or visible light. With the aid of

electronic spectra electronic state properties can be studied. However, experimen-

t alone may not be sufficient for studying excited states properties. Fortunately,

quantum mechanics methodologies can be used to complement experiment, and

help the experimentalists to make correct assignments from the spectra.

In theoretical investigations of electronic spectra, the Born-Oppenheimer ap-

proximation is assumed, that is the electrons and nuclei can be treated separately.

In this way, the electronic spectra with vibronic resolution is divided into two main

contributions; electron promotion and nuclear motion. The position of spectral

bands is determined by the transition energy and the shape of spectral profile

is determined by the band intensity, which can be calculated according to the

Franck-Condon (FC) principle.

The molecule from excited state to the ground state may emit the excessive

energy by luminescence. Based on the so-called Kasha rule [93], the fluorescence and

the phosphorescence involve maily the first singlet/triplet excited state (S1/T1)

and the singlet ground state S0, and can be expressed by,

S1 → S0 + hv(Fluorescence),

T1 → S0 + hv(Phosphorescence).(3.14)

Due to that a part of excess energy is lost via radiationless processes, the e-

mission energy is lower than the absorption energy. This energy shift, the so-called

Stockes shift, exists in absorption and emission spectra, and are schematically rep-

resented in Fig. 3.3. The characteristic lifetime of fluorescence is on a quite short

time scale (10−9 ∼10−6 s), as compared to phosphorescence where it is five orders

of magnitude larger. The lifetime of spontaneous luminescence is estimated by,

τ =1.5003

f∆E,

f =2

3∆Eµ2.

(3.15)

Here f is the dimensionless oscillator strength related to the transition dipole

moment µ.


Emission Absorption

(0,0)

(0,1)

(0,2)

(0,3)

(0,1)

(0,2)

(0,3)

Energy

Intensity

Stokes shift

(a)

01

23

Abs. Em.S0

S1

Energy

Nuclear coordinates

(b)

Em

issionA

bsorption

v=0v=1

v=2v=3

Figure 3.3 Schematic illustration of (a) Franck-Condon principle involving S1 withdominating transition (0-2), in an ideal condition of S0 with the same normal modes asS1, which leads to (a) the mirror relationship of absorption and emission spectra.

3.2.1 Fermi’s golden rule

Fermi’s golden rule describes the transition rate (transition probability per unit

time) from an initial eigenstate (Ψi) to a continuum final eigenstate (Ψf ) as a

result of a perturbation. Simply, in a time-independent interaction potential, the

Fermi’s golden rule is expressed as [94],

Wf→i =2π

~|⟨Ψf |H′|Ψi⟩|2δ(Ef − Ei). (3.16)

It is assumed that H′ is a time-dependent perturbation Hamiltonian with an

angular frequency of ω. The perturbation operator has the form [95]

H′ = Veiωt + V†e−iωt. (3.17)

Here V is time-independent operator and ω is the photon frequency.

Within the dipole approximation, the state-to-state form of Fermi’s golden

rule yields [95–97],

W stim =2π

~|⟨Ψf |µ|Ψi⟩|2δ(Ef − Ei ∓ ~ω), (3.18)

Here, the negative and positive signs correspond to the stimulated one-photon

absorption and emission, respectively.

There are two essential concepts in the description of vibrationally resolved


electronic spectra, the cross section and oscillator strength. The absorption cross

section represents the transition rate per unit photon flux, and it is written

as [96–98],

σabs(ω) ∝ ω|⟨Ψf |µ|Ψi⟩|2δ(ωif − ω), (3.19)

where ωif represents the aborpted/emitted photon frequency, ωif =Ef−Ei

~ . The

optical oscillator strength is derived from the integration over all frequencies of

Eq. (3.19) [96],

f ∝ ωif |⟨Ψf |µ|Ψi⟩|2. (3.20)

Additionally, fluorescence belongs to the spontaneous emission, whose tran-

sition rate is represented as [99],

W spont =4

3~(ωif

c)|⟨Ψf |µ|Ψi⟩|2, (3.21)

where the emission intensity is proportional to the cube of ωif ,

Ispont ∝ ω3if |⟨Ψf |µ|Ψi⟩|2. (3.22)

3.2.2 Franck-Condon principle

The Franck-Condon principle states that an intensity of vibronic transition from

one vibronic energy level to another is related to the overlap between their vibra-

tional wave functions [100]. Classically, the vertical transition originating from the

lowest vibronic level of the equilibrium geometry of the ground state to a vibronic

level of an excited state without the geometrical changes is responsible for the

peak with a significant intensity in vibronically resolved spectra [101], as shown in

Fig. 3.3. The final state is named as the FC state. According to Eq. (3.20), the

intensity of the vibronic transition is proportional to the square of the transition

dipole moment given by [6,102],

I ∝ |⟨Ψf |µ|Ψi⟩|2, (3.23)

µ = −e∑i

ri + e∑j

ZjRj = µe + µN . (3.24)

Here µ represents the dipole moment operator. r and R stand for the electron

and nuclear coordinates, respectively, and Zj the nuclear charges. Within the

BO-approximation [4], the overall wave function is a product of the electronic and

the nuclear wave functions (Ψ = ΨeΨN), where Ψe consists of electronic space and


spin wave functions and ΨN is separated into transitional (Ψt), rotational (Ψr),

and vibrational (Ψv) modes which are represented by a series of eigenfunctions of

harmonic oscillators within the harmonic approximation. Accordingly, transition

dipole moment arising from a vibronic transition is given by [103],

µi→f = ⟨Ψ′eΨ

′N |µe + µN |ΨeΨN⟩

=

∫Ψ′∗

e µe(Q)Ψedτe

∫Ψ′∗

NΨNdτv

=

∫ψ′∗e µeψedτe

∫ψ′∗s ψsdτs

∫Ψ′∗

v Ψvdτv.

(3.25)

The allowed vibronic transitions satisfy the condition that µi→f is non-zero. There-

fore, the three integrals in the Eq. (3.25) should have non-vanishing values.∫ψ′∗e µeψedτe

and∫ψ′∗s ψsdτs are named by orbital and spin selection rules, and the third inte-

gral, ⟨Ψ′∗v |Ψv⟩ is called the FC overlap. The FC factor is defined by the square

of the FC overlap [104]. There is no selection rules for the vibrational quantum

numbers, due to the non-orthogonal properties of two different electronic states

with vibronic energy levels v and v′.

The critical issue in theoretical simulations of vibrationally resolved absorp-

tion and emission spectra is the calculation of FC factors [105]. Normally, the

vibrational normal modes between the ground and excited state are not identical

overall. Therefore, the Duschinsky transformation connecting two normal modes

of ground and excited states is adapted to well describe this situation as given

by [106–109],

Q′ = JQ+K,

J = LL′T ,

K = LM1/2∆R.

(3.26)

In the above expression, Q and Q′ denote the vibrational normal coordinates of

the ground and excited states, respectively. J is named as the Duschinsky rotation

and L and L′ represent the mass-weighted normal modes in Cartesian coordinates

of the ground and excited states, respectively. M is the diagonal matrix of atomic

masses and ∆R is the displacement of two equilibrium geometries. In the limit

that the normal modes of two electron states are completely identical, J turns out

to be the identity matrix, leading to simplified calculations of FC factors. This

simplified model is called the linear coupling model (LCM). By using LCM, the

vibrationally resolved absorption and emission spectra have a mirror-symmetry


relationship around the 0-0 band, as schematically shown in Fig. 3.3.

Chapter 4

Photochemical processes

When a molecule absorbs photon energy, successive photophysical and photochem-

ical processes can take place [110]. A molecule in a high-energy level of an excited

state releases the excess energy to revert to the ground state. The whole process

is called excited-state deactivation, which can be classified as two types: chemical

and physical deactivation [111].

Photophysical deactivation does not yield new molecules, and can be divided

into intramolecular and intermolecular deactivation, according to the type of en-

ergy transfer process. Intramolecular deactivation is an energy relaxation process

by two means: radiative transition and radiationless decay. The two essential pro-

cesses differ by the transformations of electronic energy: the former emits energy

by photos and the latter converts the excessive electronic energy into rotation-

al/vibrational energy. As mentioned in the Section 3.2, the radiative transitions

include the emissions of fluorescence (F) and phosphorescence (P), arising from

the preservation and alteration of the spin state occurs along with the electron-

ic transition, respectively. On the other hand, a ‘hot’ molecule in an unstable

excited state undergoes radiationless decay involving intramolecular vibrational

relaxation (VR), internal conversion (IC) and intersystem crossing (ISC) [112]. VR

is a process where the energy transfers from high-energy to low-energy vibronic

states. IC process is featured by the electronic-state conversion with spin-state

preservation. On the contrary, the ISC process leads to a change of the spin s-

tate [113]. Owing to the spin-orbit coupling originating from the coupling of the

electron spin with the orbital angular momentum, the spin-forbidden decay is al-

lowed. IC and ISC processes may take place in the region of surface crossings.

Accordingly, the study of radiative decay means to explore the nature of surface

intersections and electronic state properties. All the photophysical intermolecu-

33

34 4 Photochemical processes

lar deactivation processes are described by the Jablonski diagram, as shown in

Fig. 4.1.

Absorption

Fluorescence

Phosphorescence

Internal conversion

Intersystem crossing

Virationalrelaxation

S1S3

T1

Nulear coordinates

v=0

v=3v=4

v=1v=2

12

34

51

23

1234

0

45

0

12

3

0

0

S4

S0

Figure 4.1 Jablonski diagram.

A competing deactivation process a ‘hot’ molecule undergoes, another way to

release the extra energy, is chemical deactivation. This involves bond cleavage and

formation, molecular rearrangement and isomerization yielding new molecules.

Taking a simple example to illustrate chemical deactivation, consider Fig. 4.2.

The reactant (R) in the ground state is promoted to the excited state (R∗), the

following reaction would proceed either via a surface intersection back to the

4.1 Potential energy surface 35

ground state yields a product (P) (as the left side of Fig. 4.2 shows), or on the

excited-state potential energy surface to the formation of product (P′∗), which

decays to P′ by emission.

RP P'

P'*

R*

S0

S1

Conicalhv

hv

intersection

Figure 4.2 Schematic view of the photochemical reaction pathways.

4.1 Potential energy surface

A correct description of the PES along the reaction coordinates is critical to in-

vestigate the mechanism of a photochemical reaction. For an N -atomic system,

the ‘real’ PES is 3N -6 (N > 2) dimensional. It is difficult to represent the reac-

tion pathway of a polyatomic molecule. In actual computations, a popular and

favorable way to describe a chemical reaction is to construct a pathway along

the reaction coordinates, which are determined by the changes in structural pa-

rameters of the key species of the reaction. This method is called the pathway

approach, suggested by Fuss et al [114]. For the ground-state reaction, these crtical

points refer to reactant, product and a transition state connecting the reactant

and the product, and the ground-state PES where the reaction takes place does

not interact with any other electronic state PESs. Accordingly, under the BO

approximation, the PES is called an adiabatic PES, which is schematically repre-

sented Fig. 4.3. In comparison to a thermal reaction, the photochemical reaction

pathway on the excited-state PES is much more complicated to describe, due to

the interplay of excited-state and ground-state PESs. The reactant, product and

transition states are not adequate to describe the photochemical reaction pathway

along the reaction coordinates, as the surface intersection leading to the electronic


state conversion via a radiationless decay process plays a critical role in the photo-

induced reaction. The surface intersection actually stands for a crossing of PESs

in terms of non-adiabatic surface where the BO approximation breaks down (see

the left side of Fig. 4.2). However, surface intersection does not always represent a

real crossing between two PESs, for instance, the avoided crossing which is typical

of adiabatic PES, as shown in Fig. 4.3. The transition rate of radiationless decay

occurring in the vicinity of real surface intersection is on a timescale of 10−15 to

10−12 s, which could be estimated by the non-adiabatic transition state theory. In

the region of an avoided crossing surface intersection, the transition rate is quite

slow as compared to the decay at a real surface intersection. In this case, the rate

constant is determined by the Franck-Condon factors and density of vibrational

states, which are predicted by Fermi’s golden rule as mentioned previously.

P

P'*hv

hv

R*

R

Avoidedcrossing

S0

S1

Figure 4.3 Schematic presentation of the reaction pathways along reaction coordinateson adiabatic PESs.

4.2 Surface intersection

Since essential nonadiabatic processes are related to PES intersection which gov-

erns the photo-induced reaction pathway, the search for a surface intersection is

a central task to investigate the photochemical reaction mechanism. As early as

1930s, von Neumann and Wigner [115] proposed the existence of conical intersection

(CI) between two PESs of electronic states from a mathematical point of view.

However, it was not employed in the explanation of photochemical reaction mech-

anisms at that time. In 1969, Teller [116] pointed out that the noncrossing rule of

4.2 Surface intersection 37

diatomics is invalid to a polyatomic molecule, which means two electronic states

could intersect in CI even if they have the same symmetry and spin multiplicity.

Zimmerman [117] and Michl [118–120] independently stated IC process takes place in

the region of CI leading to the photoproduct formation, and they first used ‘fun-

nel’ to describe the characteristic of CI. In quantum mechanics, the intersecting

adiabatic electronic state (Ψ) could be represented by the linear combination of

two orthonormal diabatic states (ϕ1 and ϕ2)[121].

Ψ = C1ϕ1 + C2ϕ2. (4.1)

According to the variational principle, the secular equation is derived [122],[H11 − E H12

H21 H22 − E

]×

[C1

C2

]= 0, (4.2)

where, H11 = ⟨ϕ1|H|ϕ1⟩, H22 = ⟨ϕ2|H|ϕ2⟩ and H12 = H21 = ⟨ϕ1|H|ϕ2⟩. Derived

from the secular equation, the energies of two states are given by [123],

E1 =1

2

[(H11 +H22)−

√(H11 −H22)2 + 4H2

12

],

E2 =1

2

[(H11 +H22) +

√(H11 −H22)2 + 4H2

12

].

(4.3)

At a surface intersection where E1=E2, it is required that

H11 = H22,

H12 = H21 = 0.(4.4)

In order to satisfy the above conditions, at least two independent variable nuclear

coordinates are needed. For the diatomic molecule, there is only one variable (i.e.

the distance of atoms). Therefore, H11=H22 is satisfied at an appropriate value

of variable, and the condition of H12=H21=0 requires that the symmetries of two

states differ by either the spatial or the spin. However, for a N -atomic system

(N>2), it is feasible to satisfy these conditions simultaneously.

At the origin where H11 = H22 = W is satisfied and H12 = H21 = 0, we use

two independent coordinates denoted as x1 and x2 and three parameters (m, n

and l) to rewrite secular equation in the first-order approximation [122,124],[W + (m+ n)x1 − E lx2

lx2 W + (m− n)x1 − E

]×

[C1

C2

]= 0, (4.5)


where,

m =1

2(h1 + h2),

n =1

2(h1 − h2),

(4.6)

with the energy expressed as,

E =W +mx1 ±√n2x2

1 + l2x22. (4.7)

It’s a equation of an elliptic double cone, and the vertex is at the origin. In the

surface intersection, the energies of the diabatic states (H11 and H22) are equal,

accordingly, the adiabatic energies is derived from Eq. (4.3),

E1 = H11 −H12,

E2 = H11 +H12,(4.8)

evidently, the energy gap between the two states is expressed as,

△E = E2 − E1 = 2H12. (4.9)

If H12 = 0, there exists a real intersection, otherwise, the intersection is split,

namely avoided crossing. As mentioned previously, the conditions for a real cross-

ing should be satisfied in the proper values of two independent variables (x1 and

x2), which are represented mathematically as [125]

x1 =∂(E1 − E2)

∂Q, (4.10)

x2 = ⟨C1|∂H∂Q|C2⟩. (4.11)

Here H represents configuration interaction Hamiltonian, C1 and C2 are eigenve-

cotrs, Q stands for the nuclear configuration vector, x1 is the gradient difference

vector and x2 is the derivative coupling vector. In the so-called branching space

(the plane spanned by the two vectors), the degeneracy is lifted, while the ener-

gy is still degenerate in the intersection space (the rest n-2 dimensions), which

is a hyperline consisting of an infinite number of conical intersections, as shown

in the right side of Fig. 4.4. A comparison between the surface intersection and

the transition state are made to feasibly understand the critical role a surface

intersection plays in a photochemical reaction. As Fig. 4.4 shows, the transition

4.2 Surface intersection 39

RP

TS

Reaction Coordinates

(a)

M*

Energy

x1

x2

CI

P

P'

Reaction Coordinates

(b)

x1

x2,x3...xn

Energy

Figure 4.4 Schematic description of (a) transition state and (b) conical intersection.

state (TS) is a highest energy point connecting two potential wells of reactant and

product along the reaction path, which is determined by a sole vector denoted as

x1, and in the remaining (n-1)-dimensional space, it is the lowest energy point.

In the reaction kinetics, TS is a bottleneck the reaction channel go through. In

contrast, the conical intersection (CI) does not connect the reactant and product

via one reaction path, however, the intersection separates the excited-state and

the ground-state PESs along reaction path. Based on the nature of the double

cone, two or more products will be yielded, due to the different reaction pathways

in the branching space via CI, as shown in the right side of Fig. 4.4.

Taking account of the spin flip in electronic state conversion, the crossings

are divided into two types: conical intersection with (n-2)-dimensional intersec-

tion space, describing the crossing derived from two electronic state in the same

spin multiplicity, and surface crossing featured by (n-1)-dimensional intersection

hyperline as the result of the interstate coupling vector vanishing. And in this

intersection, two electronic states have the different spin multiplicities. The char-

acteristics of these two type of intersections are illustrated in Fig. 4.5. The spin-

orbit coupling (SOC), leading to the forbidden reaction and intersystem crossing

processes allowed, should be taken into account at surface crossings. In my previ-

ous work included in this thesis, the SOC constant of surface crossing between the

singlet and triplet state PESs are calculated by using a one-electron approximate

spin-orbit Hamiltonian, and its value elucidates the magnitude of energy splitting

in a surface crossing.


x1

x2

(n-2)Dintersections space

Energy

2D branching space

excited state

ground state

x1

Energy

triplet

singlet

(a) (b)

Figure 4.5 Topological surface intersections (a)(n-2)-dimensional conical intersection(b)(n-1)-dimensional surface crossing.

4.3 Nonadiabatic transition state theory

Theoretical studies on thermal rate constants of nonadiabatic reactions including

photochemical reactions or spin-forbidden reactions, where a couple of electronic

states are involved, are carried out with the aid of nonadiabatic transition state

theory. Derived from Miller’s general formula of the flux-side correlation func-

tion [126] incorporated with Zhu-Nakamura theory [127–129], the semiclassical expres-

sion of rate constants is given by [130],

k =kT

Zrh

(2π

h2β

)(3N−1)/2 ∫P |∇S(Q)|dQ · δ[S(Q)]exp(−βV [S(Q)]) (4.12)

where Q represents the mass-reduced Cartesian coordinates of N-atomic system,

S(Q) denotes the crossing seam surface, Zr and P are the partition function of

reactant and the transmission probability at a crossing point, respectively. In

our previous work, the one-dimensional model was employed to calculate the rate

constant of nonadiabatic reaction, where S(Q)=s − s0 (s0 is crossing point), ac-

cordingly, Eq. (4.12) is rewritten as [130],

k =1

2πZr

∫ ∞

0

exp(−βE)Psh(E, s0)dE, (4.13)

4.3 Nonadiabatic transition state theory 41

where we use Landau-Zener equation [131,132] to treat the hopping probability (PLZ)

from one diabatic PES to another, which is expressed as,

PLZ = exp

(−2πV 2

12

~∆F

√µ

2E

), (4.14)

where V12 is the coupling element of crossing diabatic PESs. To investigate for-

bidden reactions between singlet and triplet electronic states particularly, V12 cor-

responds to the spin-orbit coupling constant. ∆F denotes the magnitude of the

gradient difference between two intersecting PESs at the crossing point along reac-

tion coordinate, µ represents the reduced mass of a normal mode in the direction

of the reaction coordinate normal to the crossing seam. The transmission prob-

ability (Psh) is closely related to PLZ . Depending on the shape of the crossing

point, the expressions of Psh are of various kinds. In normal crossing, the product

of slopes of two diabatic PESs along reaction coordinate is negative, Psh is given

by [130],

Psh =1− PLZ

1− 12PLZ

, (4.15)

On the contrary, the product is positive for an inverted crossing, Psh is expressed

as,

Psh =1− PLZ

1 + 12PLZ

. (4.16)

Incorporating with Eq. (4.13), (4.13) and (4.15) or (4.16), the rate constant of

nonadiabatic reaction is estimated in the 1D-model approximation.

Chapter 5

Summary of included papers

5.1 Molecular spectroscopy

Molecular spectroscopies are crucial implements for extracting molecular intrinsic

properties. In this thesis, I am concerned with isotropic hyperfine coupling con-

stants, which determine the shape of electron paramagnetic resonance spectrum,

and the molecular electronic spectrum. Accordingly, I make a brief summary of

my previous works on these topics.

5.1.1 Hydrogen hyperfine coupling tensor in organic rad-

icals, Papers 1-2

Density functional theory provides a useful and efficient theoretical tool to evaluate

spin Hamiltonian parameters. These works involve the application of response

theory within the framework of the spin restricted-unrestricted DFT (DFT-RU)

formalism [27,28,74], which avoids spin contamination in isotropic HFCC calculations

of π conjugated organic radicals.

We evaluated the zero-point vibrational corrections to isotropic HFCCs in

the allyl radical and four of its derivatives: the cyclobutenyl radical, the cyclopen-

tenyl radical, the bicyclo[3.1.0]hexenyl radical and the 1-hydronaphtyl radical (see

Fig. 5.1). The effect of ZPVC on the HFCCs cannot be estimated by an experi-

mental measurement directly, and what’s more, only sparse results from previous

calculations are available. Accordingly, to know how important the ZPVCs are to

HFCCs in the allyl radicals and its derivatives are, a systematical analysis is re-

quired by means of an accurate and efficient theoretical approach. We adopted the

DFT-RU approach with a strategy featured by a solving the property expanded

43

44 5 Summary of included papers

around the effective geometry as mentioned in Chapter 3 in an attempt to explore

the answers. For a good description of the spin density in the inner core and

valence region which is important for accurate HFCCs, an appropriate basis set,

HIII-su [133,134], was chosen in the HFCC calculations. According to the molecular

H

H

H

H

(1)

(2)

(3)

CHβ

C

(δ)

H2C

(1)

(2)

(3)

(m')

(p)

(m)

(o)

(β)

Allyl Cyclobutenyl Cyclopentenyl Bicyclo[3.1.0]hexenyl 1-Hydronaphthyl

(1) (3)

(1)

(2)

(3)

H(δ')

H

CH2

(1)

(2)

(3)

(β) CH2 (β)

H

HH(1)

(2)

(3)

H

H

H H

H

H

H

H

(C2V, 2A2) (C2V, 2A2) (C2V, 2A2) (Cs, 2A'') (C s, 2A'')

endo endo

exoexoHH

H

HH

H

Figure 5.1 The five allylic radicals. Selected from Paper 1, reprinted with permissionfrom The Royal Society of Chemistry.

symmetry, there are 22 unique hydrogens, where 19 hydrogens have isotropic H-

FCCs determined by the spin-polarization effects and 3 hydrogens have isotropic

HFCCs controlled by not only the spin-polarization but also the direct spin-density

contribution arising from the singly occupied molecular orbital (SOMO). Includ-

ing the ZPV contributions, not all the isotropic HFCCs are in accordance with the

experimental data [135]. It implies that the B3LYP exchange-correlation functional

systematically overestimates the isotropic HFCCs.

Based on the results from theoretical predictions, we summarized a set of

rules to guide how to make an estimate of vibrational contributions to HFCCs.

When the magnitude of HFCC is larger than 20 G, we do not need to consider

ZPVCs. Otherwise, they should be further investigated. When the isotropic

HFCC is smaller than 3 G and the distance between a hydrogen of interest and

SOMO is less than 2.5 A, we should consider if the HFCC are controlled by

spin polarization effects or the combined contributions from direct spin-density

and spin-polarization, and identify which vibrational modes make the significant

contributions.

The McConnell relation is used to describe spin density distributions in the

carbon backbone of organic π radicals from the hydrogen HFCCs measured by

electron paramagnetic resonance experiments. It is represented by ρ = QaH ,

whereQ is an empirical spin-transfer parameter given by -22.5 G [136–138]. Although

the McConnell relation is widely used in the analysis of EPR data, its origin

has not been investigated by using modern quantum chemistry. Accordingly, we

further explore the physical origin and limitations of the McConnell relation in

Paper 2. The vibrational corrections to carbon HFCCs influence not only the

5.1 Molecular spectroscopy 45

magnitude but also the sign. The model molecules chosen in our studies are

schematically plotted in Fig. 5.2. The calculations indicate that the ZPVCs

O

O

O

O

1

2

1

23

12

3

p-Benzoquinone anion

Fulvalene anion

o-Benzoquinone anion

CH2

O

CH3

1

23

4

methyl Benzyl

Tropone anion

Figure 5.2 Organic π radicals for which isotropic carbon and hydrogen HFCCs hasbeen computed with zero-point vibrational corrections.

(a) (b)

Figure 5.3 Dependence between hydrogen and carbon HFCCs in organic π radical an-ions: (a) computed at equilibrium geometry (b) with zero-point vibrational correctionsadded. Greenand blue denotes C2 positions for which in p-benzoquinone and fulvaleneanions McConnell relation predict different sign for the carbon HFCC.

overall improve the accuracy of carbon HFCCs as compared with the experimental

data. In particular, the ZPV contribution changes the sign of HFCCs of the C2

atom in p-benzoquinone anion. It suggests that the static picture is invalid for the

description of spin density and the dynamic picture with the coupling of nuclear

motion with spin density should be considered. In our studied molecules, the

non-varnishing HFCC of hydrogen atom is ascribed to the spin polarization which

leads to a rather small spin transfer constant Q and a disagreement between

the computed result and the value based on the McConnell relation. As shown


in Fig. 5.3, the linear dependence of the spin density of hydrogen and carbon

atoms represented by HFCCs is deficient without ZPVCs. In consideration of

the effects, the correct behavior is restored, especially for p-benzoquinone anion.

Thus the coupling between spin density and vibrational degrees of freedom in

the backbone is important and holds the key for understanding the spin density

transfer mechanism in such systems.

5.1.2 Vibronically resolved spectroscopy of α-santonin deriva-

tives, Paper 3

α-Santonin was the organic molecule that was first found to have a photo-induced

activity. Many experimentalists carried out systematical studies on α-santonin

photorearrangement reactions [139–141]. Recently, Natarjan and his co-workers i-

dentified the intermediates and products in solid-state reaction, moreover, record-

ed the changes of molecular spectra with the time increase [142]. We carried out

computational simulations of vibronically resolved spectra to investigate the spec-

tral characters of the molecules formed in the reaction (see Fig. 5.4), in order to

well understand the molecular spectra observed in the experiment. The com-

O

O

O

O

OO

1

2 3

45

INT-a

O 1

2 3

45

O

O

O

O

OO

H

H

12

3

4 5

P-a-I

O

O

O

OO

H

H

O

1

2

3

4 5

P-a-II

hv

OO

66

6

6

4π+2π

hv 2π+2π

O

O

O

O

OO

H

H

O

O

O

OO

H

H

O

123

4 5

1

23

4 5

D-a-II

hv

D-a-I

6

6

4π+2π

hv 2π+2π

INT-b

1' 2'

3'4'

5'6'

1' 2'

3'4'

5'

6'

1'

2'

3'4'5'

6' 1' 2'3'

4'5'

6'

santonin

Figure 5.4 Intermediates and products derived from α-santonin reactions.

putational results show that the emission energies of photosantonic acid and the

topochemical product are within the visible light range. Consequently, we studied

the vibronic spectra of photosantonic acid and the topochemical product by using

the Franck-Condon principle. The absorption spectrum of photosantonic acid has

the vibrationally resolved character consistent with experimental observations, as

5.2 Nonadiabatic reactions 47

1.5 2.0 2.5 3.0 3.5 4.0

Energy (eV)

Intensity (arb. units)

Exp.

CAS

TDDFT

Emission Absorption

Figure 5.5 Vibronically resolved spectra of photosantonic acid simulated at the CASS-CF and TDDFT levels in comparison with the experimental spectra.

shown in Fig. 5.5. Moreover, the positions of the emission and absorption bands

well match with the experimental spectra. However, the absorption band of the

topochemical product without vibrational resolution deviates considerably from

the experiment, indicating that it is not responsible for the spectra measured by

Natarjan et al [142].

5.2 Nonadiabatic reactions

Compared with adiabatic reactions, nonadiabatic reactions are more complicat-

ed, due to the involvement of surface intersections in the multiple electronic-state

PESs and the competitions in a diversity of products along the various reaction

channels. The challenges of exploring a nonadiabatic reaction mechanism are to

search for the surface intersection playing a key role along reaction pathway and

further study its character governing the radiationless decay channels, and all

attempts require the appropriate calculation method and sufficient computer re-

sources. Fortunately, with the rapid development of computational methodology

and technology in recent years, the theoretical studies on nonadiabatic reactions


make exciting achievements, which provide reasonable explanations and predic-

tions to experimental observations. In this section, I will give a brief summary of

my previous studies on the photochemical reactions originating from α-santonin

and forbidden reaction between cofactor-free enzyme and triplet oxygen molecule

with the aid of quantum mechanics incorporated with reaction kinetics.

5.2.1 α-Santonin photo-induced reactions, Papers 4-5

Under UV irradiation, α-Santonin undergoes various photorearrangement reac-

tions [143,144]. Since Kahler [145] and Alms [146] successfully isolated α-santonin in

1830s, the experimental explorations of reaction mechanisms [147,148,148–153] have

never ceased. The early works were only concerned with phenomena happening

in a photoreaction, due to limited knowledge about photochemistry and imma-

ture measurement technology. The structure of photosantonic acid in aqueous

ethanol was not reported by van Tamelen et al [154,155] until 1958. Consequent-

ly, the structures of several intermediates appearing in solution reactions were

identified, and gradually the reaction pathways were approximately estimated in

experiment [141,156,157]. Compared with the reaction taking place in solution, the

work on solid-state reaction of α-santonin progressed slowly. In 1968, a metastable

intermediate appeared and a cycloadditional product were found in solid state [158].

The critical intermediate, called cyclopentadienone, was characterized by Garcia-

Garibay et al in 2007 [142].

The photo-induced reaction pathways derived from α-santonin are illustrated

schematically in Fig. 5.6. α-Santonin undergoes a ring closed reaction leading to a

intermediate designated as INT, the following reactions face two choices: C4-C5

or C3-C4 bond cleavages, respectively. It is proposed that C4-C5 bond breaking is

feasible in solid state, due to less hindrance arising from steric effects, while C3-C4

bond breaking is favorable in solvent reaction. If the reaction starting from the

rupture of the C4-C5 bond, designated by path A [141,157], it leads to the formation

of the intermediate, A-INT. The subsequent reaction is featured by 1,2-hydrogen

immigration. Both C2 and C4 are suitable to accommodating the hydrogen atom.

The C4-H bond formation yields cyclopentadienone, labeled as A-P-I, and the

other way leads to photosantoic acid (A-P-II) [159]. As the reaction originates from

the cleavage of C3-C4 bond, named by path B [139,140], an intermediate (B-INT)

is generated. B-INT undergoing a configurational rotation of cyclopropylcarbinyl

yields lumisantonin, which can be irradiated by UV light leading to the succussive

photorearrangement reactions [139,140,157,160–167]. Lumisantonin undergoes a bond


cleavage taking place in three-membered ring, resulting in two possible reaction

pathways, denoted as path C and path D, respectively. Along path C, the reaction

generates an intermediate via the cleavage of C5-C6, namely C-INT, and the sub-

sequent reaction with a characteristic of methyl migration yields a product (C-P).

Along path D, the reaction undergoing a breakage of C4-C5 bond leads to an in-

termediate(D-INT) formation. The following reaction featured by intramolecular

hydrogen immigration generates another intermediate, labeled as D-INT-I. Due

to its enol configuration, a ketonization reaction is favorably taken place yielding

an isomer of pyrolumisantonin, namely, D-P-I. And the ratio of C-P to D-P-I

is 8:1 estimated by the experiment [162]. The reaction pathways provide a general

OO

OO

O

O

H

SantoninA-P-I

O

O

OO

O

O

H

OO

O

12

3

4

56

A-INT

A-P-II

solu

tion

O

O

O

INT B-INT

Path A Path B

O

O

OO

O

O O

O

O

OO

O

O

H2C

O

HO

O

H2C

O

O

C DPath C

Path D

hv

hv

methyl

migration

hydrogen

migrationketonization

Lumisantonin

D-INT D-INT-I D-P-I

C-INT C-P

AB

hv

Figure 5.6 Photochemical reaction pathways of α-santonin and lumisantonin suggest-ed by experiments.

picture of α-santonin photorearrangement. However, there still exist several ques-

tions. Why does the reaction depend on media? How many excited states are

involved in the photorearrangement under the experimental condition, and which

excited state governs the reaction? How does the reaction decay along radiation-

less channel via surface crossing between excited-state and ground-state PESs?

Moreover, are there any other reaction paths except for those from experimental

suggestions? All these questions are troublesome to solve by experiments; while

theoretical calculations can fill the gaps [168]. Based on the results from theoretical

calculations, we found that the 1(nπ) state is the solely accessible singlet-excited

state under the given wavelength of irradiation light. Subsequently, the 1(nπ)

state decays to the low-lying 3(ππ∗) state via a surface crossing 1(nπ∗)/3(ππ∗)-

SC-1 with a remarkable SOC as large as 55.70 cm−1 in the FC region leading to

the initiation of photorearrangement. The excited-state radiationless decays of α-

santonin in the FC region are illustrated in Fig. 5.7. The initial reaction from the

C3-C5 bond form taking place on 3(ππ∗) state PES leads to INT, where around

a surface crossing (3(ππ∗)/S0-SC-1) exists, indicating the possible transition from3(ππ∗) to the ground state. The subsequent reaction pathway is divided into two


3(ππ∗ )1(nπ∗ )

3(nπ∗ )

3(nπ∗ /ππ∗ )-CI

1(nπ∗ )/3(ππ∗ )-SC-2

3(ππ∗ )min1(nπ∗ )min

82.1

81.3

75.6

70.4~72.8

71.1~72.9 72.772.6

hv

~320nm

~89.3

IC

ISCISC

IC

ISC

VR

128.41(ππ∗ )

0.0

S0

ISCISC

ISC

ISC

84.482.1

74.7

0.0

80.1~80.7

70.8~72.0

71.0~74.0

70.4

70.3

74.1

3(ππ∗ )

3(nπ∗ )

S0

1(nπ∗ )

3(nπ∗ )min3(ππ∗ )min

1(nπ∗ )min

3(nπ∗ /ππ∗ )-CI 1(nπ∗ )/3(nπ∗ )-SC-1

1(nπ∗ )/3(ππ∗ )-SC

~95.3

hv~300nm

IC

IC

(a)

(b)

Figure 5.7 The Franck-Condon region of (a)α-santonin (b)lumisantonin.

channels. Path A is more favorable in a solid-state reaction not only due to the

involvement of the planar intermediate with a reduced space requirement but also

the advantage of the dynamic performance on the 3(ππ∗) state PES. The cleavage

of C4-C5 bond undergoing a transition state (A-TS) leads to the formation of

A-INT, and in the vicinity of it, there is a surface crossing 3(ππ∗)/S0-A-SC. If

the SC point is accessible, the reaction prefers yielding A-P-II on the ground

state PES in thermodynamics and dynamics, compared to the reaction path in

respect of A-P-I. In the dioxane reaction, path B is the preferential ascribed to

the thermodynamical preference of the biradical B-INT and lower energy barrier

leading to C3-C4 bond breakage on both PESs as well as smaller energy barrier

overcome resulting in lumisantonin formation on ground-state PES, in comparison

with reactions along path A. Around B-INT, a state conversion would happen

via a surface crossing (3(ππ∗)/S0-B-SC), and finally lumisantonin is yielded via


almost barrier free process. Exposed to UV light with a wavelength of 320nm,

lumisantonin is excited to 1(nπ∗) state with a notable oscillator strength and sub-

sequently decays to 3(ππ∗) state via a surface crossing (1(nπ∗)/3(ππ∗)-SC-2) in the

FC region. The photophysical processes are quite analogous to the decay channels

of α-santonin in the FC region. Interestingly, the 3(ππ∗) state originates from the

electronic promotion from π-type orbital featured by a mixture of π-type orbital

coupling with σ-type orbital of C-C bond in the three-membered carbon ring to

π∗ orbital. Due to the nature of the 3(ππ∗) state, the C-C bond breaks. Along

path C, starting from C5-C6 bond cleavage, the reaction undergoes a barrier-free

channel on the ground- and 3(ππ∗)-state PESs. Particularly, it is preferential

both dynamically and thermodynamically on 3(ππ∗)-state PES. In the vicinity of

C-INT, a surface crossing (3(ππ∗)/S0-C-SC) makes the deactivation decay from3(ππ∗) to the ground state possible, leading to the recovery of B-P with C5-C6

bond reformed in the ground state. Otherwise, an energy barrier should be over-

come to fulfil the reaction yielding C-P in the 3(ππ∗) state. Along path D, arising

from C5-C4 bond rupture, the reaction is characteristic of a concerted reaction

consisting of two stages: bond rupture and a hydrogen or methyl migration on

the ground-state PES. The reaction involving a hydrogen shift is labeled as path

D-I and a methyl migration is called as path D-II. Compared with the reaction

along path C, the channel proceeding along path D on the ground state PES is

more favorable due to a stable intermediate separated from B-P by a transition

state. Moreover, the energy barrier corresponding to the bond rupture and hy-

drogen shift is much lower than that of the reaction assigned to bond rupture and

methyl migration. Accordingly, the formation of D-INT-I is overwhelming on

the ground state PES. On the 3(ππ∗) state PES, a surface crossing (3(ππ∗)/S0-D-

SC) adjacent to the D-INT indicates the intersystem crossing may take place.

The successive stage should surmount much higher energy barrier to complete

the reaction than its counterpart in path C. Overall, photolytic reaction along

path C is more favorable dynamically and thermodynamically on the 3(ππ∗)-state

PES, while the pyrolytic reaction along path D is facile owing to a stable inter-

mediate generated. These results provide a further insight into the experimental

observations.


5.2.2 Reaction of Cofactor-Free Enzyme with Triplet Oxy-

gen Molecule, Paper 6

Oxidation reactions catalyzed by a wide variety of enzymes are the most common

chemical reactions occurring in the primary and secondary metabolism of organ-

isms. Many oxygenation enzymes with the help of transition metal ions or organic

cofactors active substrate and/or dioxygen [169,170]. There are some exceptions that

they are independent of cofactors, such as the quinone-forming monooxygenases,

the bacterial dioxygenases and the vancomyclin biosynthetic dioxygenases [171–173].

For cofactor-free dioxygenase, the amino acid residue plays a key role in the ox-

idation catalyzed reaction. It is a ‘spin-forbidden’ reaction, due to the interplay

of triplet-state dioxygen and singlet-state substrate. Accordingly, how the spin

state of dioxygen changes is an essential question concerned with the reaction

mechanism. As experimentalists suggested, the common mechanism of oxidation

reaction catalyzed by cofactor-independent enzyme could be described that a pro-

ton of substrate is firstly abstracted by an amino acid residue, subsequently, the

anionic substrate feasibly transfers one electron to oxygen molecule leading to

the formation of caged radical pair [169], which is a critical precursor for the se-

quent reaction. The proposed general reaction mechanism is derived from crystal

structure, however, the static picture is limited in description of the underly-

ing reaction mechanism. The theoretical studies can provide useful information

about the electronic structures and the reaction energies. In this paper, we regard

the vancomycin biosynthetic enzyme DpgC as the original model to study the

spin-forbidden reaction. We expect the calculations will provide an insight into

understanding spin flip of dioxygen in the oxidation.

As shown in Fig. 5.8, we design three models under investigation: anionic

sulfidomethyl propanethioate, anionic sulfidomethyl but-3-enethioate and anionic

sulfidomethyl 2-(3,5-dihydroxyphenyl) ethanethioate, respectively, for the explo-

ration in the substituent effects on the reaction mechanism. The theoretical

studies show that the electron-transfer mechanism is not a sole way to make

the spin-forbidden oxidation allowed. Only the reaction of anionic sulfidomethyl

propanethioate with dioxygen shows a significant charge migration, and the rest

models obey another reaction mechanism, which involves the surface hopping be-

tween the singlet- and triplet-state PESs via surface crossing, where the electronic

configuration is mixed with the singlet- and triplet- state character, and no sig-

nificant charge migration occurs from the substrate to the molecular oxygen.

As shown in fig. 5.9. The frontier orbitals of (S/T)SC-I are quite similar


[3O2 + CH3Cα HCOSCH3]- 1[CH3HCα CHOSCH3]-

[3O2 + C2H3Cα HCOSCH3]- 1[C2H3HCα CHOSCH3]-

O O

[3O2 + C6H6O2Cα HCOSCH3]- 1[C6H6O2Cα HCOSCH3]-

I.

II.

III.

O O

O O

Figure 5.8 Oxidation reaction of the computational models from the reactants to thefist intermediates.

(a) (c)(b)

1.116 0.910 1.490 0.612 1.500 0.621

Figure 5.9 Schematic view of molecular orbtials and the electron occupancies of min-imum energy crossing points (a) (S/T)SC-I, (b) (S/T)SC-II and (c) (S/T)SC-III.

except that the orbital phases on the substrate are reversed. The electron occu-

pancies are 1.116 and 0.910, respectively, indicating that (S/T)SC-I is a biradical.

In contrast, the plotted frontier orbitals of (S/T)SC-II are quite different, one is

dioxygen π∗ orbital with the electron occupancies of 1.490, and the other is π-type

orbital localized on the dioxygen and the substrate with the electron occupancies

of 0.612. The character of the frontier orbitals of (S/T)SC-III has some analogy

to that of (S/T)SC-II. The electron occupancies are shown as 1.500 and 0.621

in these two orbitals, respectively. It should be mentioned that the newly formed

O-Cα bond removes the degenerate highest occupied molecular orbitals localized

on the dioxygen. However, the interaction between the π-type frontier orbital-

s localized in the conjugated substituents (i.e. vinyl and resorcinol) and the π∗

orbital of dioxygen gives rise to the dioxygen away from the substrates.

The conjugated substrate can stabilize the dioxygen at a long distance away

from it, and the charge-transfer mechanism does not manipulate this kind of

oxidation. With the size of conjugated substrate increasing, the SOC constant at

a minimum of energy crossing point gets larger, indicating the interplay between

the single- and triplet-state PESs is enhanced. However, the rate of the spin-flip

process becomes lower and determines the whole stage rate of the oxidation, since

the energy barrier of the spin-state conversion is increased.

The energies of an α-hydrogen abstracted from the substrates indicate that

the non-conjugated conformation is relatively unfavorable to the loss of a proton


from α-carbon, however, the conjugated substrate is propitious for the stabiliza-

tion of the negative charge after a hydrogen abstracted, which is cardinal reason

why the spin flip appears without the significant charge migration. The design of

cofactor-free enzyme with a good performance is required to take all factors into

consideration. Generally speaking, a strongly conjugated cofactor-free substrate

is in favor of the oxidation catalysis.

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